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}

     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 \label{sec:numerals}

    39

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

    41 and reals and denote integer values of arbitrary size.

    42 \REMARK{hypreal?}

    43 They begin

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

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

    46 and \isa{\#441223334678}.

    47

    48 Literals look like constants, but they abbreviate

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

    50 Isabelle performs arithmetic on literals by rewriting, rather

    51 than using the hardware arithmetic. In most cases arithmetic

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

    53 provided for literals are addition, subtraction, multiplication,

    54 integer division and remainder.

    55

    56 \emph{Beware}: the arithmetic operators are

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

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

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

    60 \begin{isabelle}

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

    62 \end{isabelle}

    63 %

    64 Carefully observe how Isabelle displays the subgoal:

    65 \begin{isabelle}

    66 \ 1.\ (\#2::'a)\ *\ m\ =\ m\ +\ m

    67 \end{isabelle}

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

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

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

    71

    72

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

    74

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

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

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

    78 in exceptional circumstances should you resort to induction.

    79

    80 \subsubsection{Literals}

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

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

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

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

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

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

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

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

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

    90 primitive recursive functions. The following default  simplification rules

    91 replace small literals by zero and successor:

    92 \begin{isabelle}

    93 \#0\ =\ 0

    94 \rulename{numeral_0_eq_0}\isanewline

    95 \#1\ =\ 1

    96 \rulename{numeral_1_eq_1}\isanewline

    97 \#2\ +\ n\ =\ Suc\ (Suc\ n)

    98 \rulename{add_2_eq_Suc}\isanewline

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

   100 \rulename{add_2_eq_Suc'}

   101 \end{isabelle}

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

   103 these rules.

   104

   105 \subsubsection{Typical lemmas}

   106 Inequalities involving addition and subtraction alone can be proved

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

   108 involving multiplication and division:

   109 \begin{isabelle}

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

   111 \rulename{mult_le_mono}\isanewline

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

   113 *\ k\ <\ j\ *\ k%

   114 \rulename{mult_less_mono1}\isanewline

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

   116 \rulename{div_le_mono}

   117 \end{isabelle}

   118 %

   119 Various distributive laws concerning multiplication are available:

   120 \begin{isabelle}

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

   122 \rulename{add_mult_distrib}\isanewline

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

   124 \rulename{diff_mult_distrib}\isanewline

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

   126 \rulename{mod_mult_distrib}

   127 \end{isabelle}

   128

   129 \subsubsection{Division}

   130 The library contains the basic facts about quotient and remainder

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

   132 \begin{isabelle}

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

   134 \rulename{mod_if}\isanewline

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

   136 \rulename{mod_div_equality}

   137 \end{isabelle}

   138

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

   140 Here is a selection:

   141 \begin{isabelle}

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

   143 \rulename{div_mult1_eq}\isanewline

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

   145 \rulename{mod_mult1_eq}\isanewline

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

   147 \rulename{div_mult2_eq}\isanewline

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

   149 \rulename{mod_mult2_eq}\isanewline

   150 0\ <\ c\ \isasymLongrightarrow \ (c\ *\ a)\ div\ (c\ *\ b)\ =\ a\ div\ b%

   151 \rulename{div_mult_mult1}

   152 \end{isabelle}

   153

   154 Surprisingly few of these results depend upon the

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

   156 \begin{isabelle}

   157 a\ div\ 0\ =\ 0

   158 \rulename{DIVISION_BY_ZERO_DIV}\isanewline

   159 a\ mod\ 0\ =\ a%

   160 \rulename{DIVISION_BY_ZERO_MOD}

   161 \end{isabelle}

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

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

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

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

   166

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

   168 is overloaded over all numeric types:

   169 \begin{isabelle}

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

   171 \rulename{dvd_def}

   172 \end{isabelle}

   173 %

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

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

   176 \begin{isabelle}

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

   178 \rulename{dvd_anti_sym}\isanewline

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

   180 \rulename{dvd_add}

   181 \end{isabelle}

   182

   183 \subsubsection{Simplifier tricks}

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

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

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

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

   188 using this split rule:

   189 \begin{isabelle}

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

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

   192 d))

   193 \rulename{nat_diff_split}

   194 \end{isabelle}

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

   196 linear arithmetic:

   197 \begin{isabelle}

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

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

   200 \isacommand{done}

   201 \end{isabelle}

   202

   203 Suppose that two expressions are equal, differing only in

   204 associativity and commutativity of addition.  Simplifying with the

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

   206 the two expressions identical:

   207 \begin{isabelle}

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

   209 \rulename{add_assoc}\isanewline

   210 m\ +\ n\ =\ n\ +\ m%

   211 \rulename{add_commute}\isanewline

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

   213 +\ z)

   214 \rulename{add_left_commute}

   215 \end{isabelle}

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

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

   218 with this goal:

   219 \begin{isabelle}

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

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

   222 \end{isabelle}

   223 %

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

   225 \begin{isabelle}

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

   227 \end{isabelle}

   228 %

   229 Here is the resulting subgoal:

   230 \begin{isabelle}

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

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

   233 \end{isabelle}

   234

   235

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

   237

   238 Reasoning methods resemble those for the natural numbers, but

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

   240

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

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

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

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

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

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

   247

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

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

   250 of the divisor:

   251 \begin{isabelle}

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

   253 \rulename{pos_mod_sign}\isanewline

   254 \#0\ <\ b\ \isasymLongrightarrow \ a\ mod\ b\ <\ b%

   255 \rulename{pos_mod_bound}\isanewline

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

   257 \rulename{neg_mod_sign}\isanewline

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

   259 \rulename{neg_mod_bound}

   260 \end{isabelle}

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

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

   263

   264 The library provides many lemmas for proving inequalities involving integer

   265 multiplication and division, similar to those shown above for

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

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

   268 \begin{isabelle}

   269 \isasymbar x\ *\ y\isasymbar \ =\ \isasymbar x\isasymbar \ *\ \isasymbar y\isasymbar

   270 \rulename{abs_mult}

   271 \end{isabelle}

   272

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

   274 \begin{isabelle}

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

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

   277 \rulename{zdiv_zadd1_eq}

   278 \par\smallskip

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

   280 \rulename{zmod_zadd1_eq}

   281 \end{isabelle}

   282

   283 \begin{isabelle}

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

   285 \rulename{zdiv_zmult1_eq}\isanewline

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

   287 \rulename{zmod_zmult1_eq}

   288 \end{isabelle}

   289

   290 \begin{isabelle}

   291 \#0\ <\ c\ \isasymLongrightarrow \ a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%

   292 \rulename{zdiv_zmult2_eq}\isanewline

   293 \#0\ <\ c\ \isasymLongrightarrow \ a\ mod\ (b*c)\ =\ b*(a\ div\ b\ mod\

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

   295 \rulename{zmod_zmult2_eq}

   296 \end{isabelle}

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

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

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

   300 is

   301 $\isa{a} = 7$, $\isa{b} = 2$ and $\isa{c} = -3$, when the left-hand side of

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

   303

   304

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

   306

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

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

   309 associativity and commutativity theorems for addition; similarly

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

   311

   312 \textbf{To be written.  Inverse, abs, theorems about density, etc.?}