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doc-src/TutorialI/Types/numerics.tex

author | nipkow |

Thu Jan 04 18:13:27 2001 +0100 (2001-01-04) | |

changeset 10779 | b0d961105f46 |

parent 10777 | a5a6255748c3 |

child 10794 | 65d18005d802 |

permissions | -rw-r--r-- |

label!

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.

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.

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.

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.

32 \subsection{Numeric Literals}

33 \label{sec:numerals}

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}.

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.

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.

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

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.

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.

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}

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}

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}

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.

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}

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}

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}

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

232 Reasoning methods resemble those for the natural numbers, but

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

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.

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.

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}

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}

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}

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 $-1$.

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

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.)

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}.

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}

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}

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}

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}

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.

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}

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}.

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 $\approx$.