doc-src/TutorialI/Types/numerics.tex
author paulson
Tue Dec 05 18:55:18 2000 +0100 (2000-12-05)
changeset 10594 6330bc4b6fe4
child 10608 620647438780
permissions -rw-r--r--
nat and int sections but no real
     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}
     5 of \textbf{integers}, which gives up induction in exchange  for proper subtraction.
     6 
     7 The integers are preferable to the natural  numbers for reasoning about
     8 complicated arithmetic expressions. For  example, a termination proof
     9 typically involves an integer metric  that is shown to decrease at each
    10 loop iteration. Even if the  metric cannot become negative, proofs about it
    11 are usually easier  if the integers are used rather than the natural
    12 numbers. 
    13 
    14 The logic Isabelle/HOL-Real also has the type \isa{real} of real numbers
    15 and even the type \isa{hypreal} of non-standard reals. These
    16 \textbf{hyperreals} include  infinitesimals, which represent infinitely
    17 small and infinitely  large quantities; they greatly facilitate proofs
    18 about limits,  differentiation and integration.  Isabelle has no subtyping, 
    19 so the numeric types are distinct and there are 
    20 functions to convert between them. 
    21 
    22 Many theorems involving numeric types can be proved automatically by
    23 Isabelle's arithmetic decision procedure, the method
    24 \isa{arith}.  Linear arithmetic comprises addition, subtraction
    25 and multiplication by constant factors; subterms involving other operators
    26 are regarded as variables.  The procedure can be slow, especially if the
    27 subgoal to be proved involves subtraction over type \isa{nat}, which 
    28 causes case splits.  
    29 
    30 The simplifier reduces arithmetic expressions in other
    31 ways, such as dividing through by common factors.  For problems that lie
    32 outside the scope of automation, the library has hundreds of
    33 theorems about multiplication, division, etc., that can be brought to
    34 bear.  You can find find them by browsing the library.  Some
    35 useful lemmas are shown below.
    36 
    37 \subsection{Numeric Literals}
    38 
    39 Literals are available for the types of natural numbers, integers 
    40 and reals and denote integer values of arbitrary size. 
    41 \REMARK{hypreal?}
    42 They begin 
    43 with a number sign (\isa{\#}), have an optional minus sign (\isa{-}) and 
    44 then one or more decimal digits. Examples are \isa{\#0}, \isa{\#-3} 
    45 and \isa{\#441223334678}.
    46 
    47 Literals look like constants, but they abbreviate 
    48 terms, representing the number in a two's complement binary notation. 
    49 Isabelle performs arithmetic on literals by rewriting, rather 
    50 than using the hardware arithmetic. In most cases arithmetic 
    51 is fast enough, even for large numbers. The arithmetic operations 
    52 provided for literals are addition, subtraction, multiplication, 
    53 integer division and remainder. 
    54 
    55 \emph{Beware}: the arithmetic operators are 
    56 overloaded, so you must be careful to ensure that each numeric 
    57 expression refers to a specific type, if necessary by inserting 
    58 type constraints.  Here is an example of what can go wrong:
    59 \begin{isabelle}
    60 \isacommand{lemma}\ "\#2\ *\ m\ =\ m\ +\ m"
    61 \end{isabelle}
    62 %
    63 Carefully observe how Isabelle displays the subgoal:
    64 \begin{isabelle}
    65 \ 1.\ (\#2::'a)\ *\ m\ =\ m\ +\ m
    66 \end{isabelle}
    67 The type \isa{'a} given for the literal \isa{\#2} warns us that no numeric
    68 type has been specified.  The problem is underspecified.  Given a type
    69 constraint such as \isa{nat}, \isa{int} or \isa{real}, it becomes trivial.
    70 
    71 
    72 \subsection{The type of natural numbers, {\tt\slshape nat}}
    73 
    74 This type requires no introduction: we have been using it from the
    75 start.  Hundreds of useful lemmas about arithmetic on type \isa{nat} are
    76 proved in the theories \isa{Nat}, \isa{NatArith} and \isa{Divides}.  Only
    77 in exceptional circumstances should you resort to induction.
    78 
    79 \subsubsection{Literals}
    80 The notational options for the natural numbers can be confusing. The 
    81 constant \isa{0} is overloaded to serve as the neutral value 
    82 in a variety of additive types. The symbols \isa{1} and \isa{2} are 
    83 not constants but abbreviations for \isa{Suc 0} and \isa{Suc(Suc 0)},
    84 respectively. The literals \isa{\#0}, \isa{\#1} and \isa{\#2}  are
    85 entirely different from \isa{0}, \isa{1} and \isa{2}. You  will
    86 sometimes prefer one notation to the other. Literals are  obviously
    87 necessary to express large values, while \isa{0} and \isa{Suc}  are
    88 needed in order to match many theorems, including the rewrite  rules for
    89 primitive recursive functions. The following default  simplification rules
    90 replace small literals by zero and successor: 
    91 \begin{isabelle}
    92 \#0\ =\ 0
    93 \rulename{numeral_0_eq_0}\isanewline
    94 \#1\ =\ 1
    95 \rulename{numeral_1_eq_1}\isanewline
    96 \#2\ +\ n\ =\ Suc\ (Suc\ n)
    97 \rulename{add_2_eq_Suc}\isanewline
    98 n\ +\ \#2\ =\ Suc\ (Suc\ n)
    99 \rulename{add_2_eq_Suc'}
   100 \end{isabelle}
   101 In special circumstances, you may wish to remove or reorient 
   102 these rules. 
   103 
   104 \subsubsection{Typical lemmas}
   105 Inequalities involving addition and subtraction alone can be proved
   106 automatically.  Lemmas such as these can be used to prove inequalities
   107 involving multiplication and division:
   108 \begin{isabelle}
   109 \isasymlbrakk i\ \isasymle \ j;\ k\ \isasymle \ l\isasymrbrakk \ \isasymLongrightarrow \ i\ *\ k\ \isasymle \ j\ *\ l%
   110 \rulename{mult_le_mono}\isanewline
   111 \isasymlbrakk i\ <\ j;\ 0\ <\ k\isasymrbrakk \ \isasymLongrightarrow \ i\
   112 *\ k\ <\ j\ *\ k%
   113 \rulename{mult_less_mono1}\isanewline
   114 m\ \isasymle \ n\ \isasymLongrightarrow \ m\ div\ k\ \isasymle \ n\ div\ k%
   115 \rulename{div_le_mono}
   116 \end{isabelle}
   117 %
   118 Various distributive laws concerning multiplication are available:
   119 \begin{isabelle}
   120 (m\ +\ n)\ *\ k\ =\ m\ *\ k\ +\ n\ *\ k%
   121 \rulename{add_mult_distrib}\isanewline
   122 (m\ -\ n)\ *\ k\ =\ m\ *\ k\ -\ n\ *\ k%
   123 \rulename{diff_mult_distrib}\isanewline
   124 (m\ mod\ n)\ *\ k\ =\ (m\ *\ k)\ mod\ (n\ *\ k)
   125 \rulename{mod_mult_distrib}
   126 \end{isabelle}
   127 
   128 \subsubsection{Division}
   129 The library contains the basic facts about quotient and remainder
   130 (including the analogous equation, \isa{div_if}):
   131 \begin{isabelle}
   132 m\ mod\ n\ =\ (if\ m\ <\ n\ then\ m\ else\ (m\ -\ n)\ mod\ n)
   133 \rulename{mod_if}\isanewline
   134 m\ div\ n\ *\ n\ +\ m\ mod\ n\ =\ m%
   135 \rulename{mod_div_equality}
   136 \end{isabelle}
   137 
   138 Many less obvious facts about quotient and remainder are also provided. 
   139 Here is a selection:
   140 \begin{isabelle}
   141 a\ *\ b\ div\ c\ =\ a\ *\ (b\ div\ c)\ +\ a\ *\ (b\ mod\ c)\ div\ c%
   142 \rulename{div_mult1_eq}\isanewline
   143 a\ *\ b\ mod\ c\ =\ a\ *\ (b\ mod\ c)\ mod\ c%
   144 \rulename{mod_mult1_eq}\isanewline
   145 a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%
   146 \rulename{div_mult2_eq}\isanewline
   147 a\ mod\ (b*c)\ =\ b * (a\ div\ b\ mod\ c)\ +\ a\ mod\ b%
   148 \rulename{mod_mult2_eq}\isanewline
   149 0\ <\ c\ \isasymLongrightarrow \ (c\ *\ a)\ div\ (c\ *\ b)\ =\ a\ div\ b%
   150 \rulename{div_mult_mult1}
   151 \end{isabelle}
   152 
   153 Surprisingly few of these results depend upon the
   154 divisors' being nonzero.  Isabelle/HOL defines division by zero:
   155 \begin{isabelle}
   156 a\ div\ 0\ =\ 0
   157 \rulename{DIVISION_BY_ZERO_DIV}\isanewline
   158 a\ mod\ 0\ =\ a%
   159 \rulename{DIVISION_BY_ZERO_MOD}
   160 \end{isabelle}
   161 As a concession to convention, these equations are not installed as default
   162 simplification rules but are merely used to remove nonzero-divisor
   163 hypotheses by case analysis.  In \isa{div_mult_mult1} above, one of
   164 the two divisors (namely~\isa{c}) must be still be nonzero.
   165 
   166 The \textbf{divides} relation has the standard definition, which
   167 is overloaded over all numeric types: 
   168 \begin{isabelle}
   169 m\ dvd\ n\ \isasymequiv\ {\isasymexists}k.\ n\ =\ m\ *\ k
   170 \rulename{dvd_def}
   171 \end{isabelle}
   172 %
   173 Section~\ref{sec:proving-euclid} discusses proofs involving this
   174 relation.  Here are some of the facts proved about it:
   175 \begin{isabelle}
   176 \isasymlbrakk m\ dvd\ n;\ n\ dvd\ m\isasymrbrakk \ \isasymLongrightarrow \ m\ =\ n%
   177 \rulename{dvd_anti_sym}\isanewline
   178 \isasymlbrakk k\ dvd\ m;\ k\ dvd\ n\isasymrbrakk \ \isasymLongrightarrow \ k\ dvd\ (m\ +\ n)
   179 \rulename{dvd_add}
   180 \end{isabelle}
   181 
   182 \subsubsection{Simplifier tricks}
   183 The rule \isa{diff_mult_distrib} shown above is one of the few facts
   184 about \isa{m\ -\ n} that is not subject to
   185 the condition \isa{n\ \isasymle \  m}.  Natural number subtraction has few
   186 nice properties; often it is best to remove it from a subgoal
   187 using this split rule:
   188 \begin{isabelle}
   189 P(a-b)\ =\ ((a<b\ \isasymlongrightarrow \ P\
   190 0)\ \isasymand \ (\isasymforall d.\ a\ =\ b+d\ \isasymlongrightarrow \ P\
   191 d))
   192 \rulename{nat_diff_split}
   193 \end{isabelle}
   194 For example, it proves the following fact, which lies outside the scope of
   195 linear arithmetic:
   196 \begin{isabelle}
   197 \isacommand{lemma}\ "(n-1)*(n+1)\ =\ n*n\ -\ 1"\isanewline
   198 \isacommand{apply}\ (simp\ split:\ nat_diff_split)\isanewline
   199 \isacommand{done}
   200 \end{isabelle}
   201 
   202 Suppose that two expressions are equal, differing only in 
   203 associativity and commutativity of addition.  Simplifying with the
   204 following equations sorts the terms and groups them to the right, making
   205 the two expressions identical:
   206 \begin{isabelle}
   207 m\ +\ n\ +\ k\ =\ m\ +\ (n\ +\ k)
   208 \rulename{add_assoc}\isanewline
   209 m\ +\ n\ =\ n\ +\ m%
   210 \rulename{add_commute}\isanewline
   211 x\ +\ (y\ +\ z)\ =\ y\ +\ (x\
   212 +\ z)
   213 \rulename{add_left_commute}
   214 \end{isabelle}
   215 The name \isa{add_ac} refers to the list of all three theorems, similarly
   216 there is \isa{mult_ac}.  Here is an example of the sorting effect.  Start
   217 with this goal:
   218 \begin{isabelle}
   219 \ 1.\ Suc\ (i\ +\ j\ *\ l\ *\ k\ +\ m\ *\ n)\ =\
   220 f\ (n\ *\ m\ +\ i\ +\ k\ *\ j\ *\ l)
   221 \end{isabelle}
   222 %
   223 Simplify using  \isa{add_ac} and \isa{mult_ac}:
   224 \begin{isabelle}
   225 \isacommand{apply}\ (simp\ add:\ add_ac\ mult_ac)
   226 \end{isabelle}
   227 %
   228 Here is the resulting subgoal:
   229 \begin{isabelle}
   230 \ 1.\ Suc\ (i\ +\ (m\ *\ n\ +\ j\ *\ (k\ *\ l)))\
   231 =\ f\ (i\ +\ (m\ *\ n\ +\ j\ *\ (k\ *\ l)))%
   232 \end{isabelle}
   233 
   234 
   235 \subsection{The type of integers, {\tt\slshape int}}
   236 
   237 Reasoning methods resemble those for the natural numbers, but
   238 induction and the constant \isa{Suc} are not available.
   239 
   240 Concerning simplifier tricks, we have no need to eliminate subtraction (it
   241 is well-behaved), but the simplifier can sort the operands of integer
   242 operators.  The name \isa{zadd_ac} refers to the associativity and
   243 commutativity theorems for integer addition, while \isa{zmult_ac} has the
   244 analogous theorems for multiplication.  The prefix~\isa{z} in many theorem
   245 names recalls the use of $\Bbb{Z}$ to denote the set of integers.
   246 
   247 For division and remainder, the treatment of negative divisors follows
   248 traditional mathematical practice: the sign of the remainder follows that
   249 of the divisor:
   250 \begin{isabelle}
   251 \#0\ <\ b\ \isasymLongrightarrow \ \#0\ \isasymle \ a\ mod\ b%
   252 \rulename{pos_mod_sign}\isanewline
   253 \#0\ <\ b\ \isasymLongrightarrow \ a\ mod\ b\ <\ b%
   254 \rulename{pos_mod_bound}\isanewline
   255 b\ <\ \#0\ \isasymLongrightarrow \ a\ mod\ b\ \isasymle \ \#0
   256 \rulename{neg_mod_sign}\isanewline
   257 b\ <\ \#0\ \isasymLongrightarrow \ b\ <\ a\ mod\ b%
   258 \rulename{neg_mod_bound}
   259 \end{isabelle}
   260 ML treats negative divisors in the same way, but most computer hardware
   261 treats signed operands using the same rules as for multiplication.
   262 
   263 The library provides many lemmas for proving inequalities involving integer
   264 multiplication and division, similar to those shown above for
   265 type~\isa{nat}.  The absolute value function \isa{abs} is
   266 defined for the integers; we have for example the obvious law
   267 \begin{isabelle}
   268 \isasymbar x\ *\ y\isasymbar \ =\ \isasymbar x\isasymbar \ *\ \isasymbar y\isasymbar 
   269 \rulename{abs_mult}
   270 \end{isabelle}
   271 
   272 Again, many facts about quotients and remainders are provided:
   273 \begin{isabelle}
   274 (a\ +\ b)\ div\ c\ =\isanewline
   275 a\ div\ c\ +\ b\ div\ c\ +\ (a\ mod\ c\ +\ b\ mod\ c)\ div\ c%
   276 \rulename{zdiv_zadd1_eq}
   277 \par\smallskip
   278 (a\ +\ b)\ mod\ c\ =\ (a\ mod\ c\ +\ b\ mod\ c)\ mod\ c%
   279 \rulename{zmod_zadd1_eq}
   280 \end{isabelle}
   281 
   282 \begin{isabelle}
   283 (a\ *\ b)\ div\ c\ =\ a\ *\ (b\ div\ c)\ +\ a\ *\ (b\ mod\ c)\ div\ c%
   284 \rulename{zdiv_zmult1_eq}\isanewline
   285 (a\ *\ b)\ mod\ c\ =\ a\ *\ (b\ mod\ c)\ mod\ c%
   286 \rulename{zmod_zmult1_eq}
   287 \end{isabelle}
   288 
   289 \begin{isabelle}
   290 \#0\ <\ c\ \isasymLongrightarrow \ a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%
   291 \rulename{zdiv_zmult2_eq}\isanewline
   292 \#0\ <\ c\ \isasymLongrightarrow \ a\ mod\ (b*c)\ =\ b*(a\ div\ b\ mod\
   293 c)\ +\ a\ mod\ b%
   294 \rulename{zmod_zmult2_eq}
   295 \end{isabelle}
   296 The last two differ from their natural number analogues by requiring
   297 \isa{c} to be positive.  Since division by zero yields zero, we could allow
   298 \isa{c} to be zero.  However, \isa{c} cannot be negative: a counterexample
   299 is
   300 $\isa{a} = 7$, $\isa{b} = 2$ and $\isa{c} = -3$, when the left-hand side of
   301 \isa{zdiv_zmult2_eq} is $-2$ while the right-hand side is $-1$.
   302 
   303 
   304 \subsection{The type of real numbers, {\tt\slshape real}}
   305 
   306 As with the other numeric types, the simplifier can sort the operands of
   307 addition and multiplication.  The name \isa{real_add_ac} refers to the
   308 associativity and commutativity theorems for addition; similarly
   309 \isa{real_mult_ac} contains those properties for multiplication. 
   310 
   311 \textbf{To be written.  Inverse, abs, theorems about density, etc.?}