doc-src/TutorialI/Types/numerics.tex
author nipkow
Wed Jun 22 09:26:18 2005 +0200 (2005-06-22)
changeset 16523 f8a734dc0fbc
parent 16412 50eab0183aea
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permissions -rw-r--r--
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% $Id$
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\section{Numbers}
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\label{sec:numbers}
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\index{numbers|(}%
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Until now, our numerical examples have used the type of \textbf{natural
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numbers},
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\isa{nat}.  This is a recursive datatype generated by the constructors
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zero  and successor, so it works well with inductive proofs and primitive
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recursive function definitions.  HOL also provides the type
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\isa{int} of \textbf{integers}, which lack induction but support true
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subtraction.  With subtraction, arithmetic reasoning is easier, which makes
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the integers preferable to the natural numbers for
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complicated arithmetic expressions, even if they are non-negative.  The logic HOL-Complex also has the types
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\isa{rat}, \isa{real} and \isa{complex}: the rational, real and complex numbers.  Isabelle has no 
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subtyping,  so the numeric
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types are distinct and there are functions to convert between them.
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Most numeric operations are overloaded: the same symbol can be
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used at all numeric types. Table~\ref{tab:overloading} in the appendix
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shows the most important operations, together with the priorities of the
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infix symbols. Algebraic properties are organized using type classes
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around algebraic concepts such as rings and fields;
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a property such as the commutativity of addition is a single theorem 
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(\isa{add_commute}) that applies to all numeric types.
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\index{linear arithmetic}%
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Many theorems involving numeric types can be proved automatically by
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Isabelle's arithmetic decision procedure, the method
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\methdx{arith}.  Linear arithmetic comprises addition, subtraction
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and multiplication by constant factors; subterms involving other operators
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are regarded as variables.  The procedure can be slow, especially if the
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subgoal to be proved involves subtraction over type \isa{nat}, which 
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causes case splits.  On types \isa{nat} and \isa{int}, \methdx{arith}
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can deal with quantifiers---this is known as Presburger arithmetic---whereas on type \isa{real} it cannot.
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The simplifier reduces arithmetic expressions in other
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ways, such as dividing through by common factors.  For problems that lie
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outside the scope of automation, HOL provides hundreds of
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theorems about multiplication, division, etc., that can be brought to
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bear.  You can locate them using Proof General's Find
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button.  A few lemmas are given below to show what
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is available.
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\subsection{Numeric Literals}
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\label{sec:numerals}
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\index{numeric literals|(}%
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The constants \cdx{0} and \cdx{1} are overloaded.  They denote zero and one,
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respectively, for all numeric types.  Other values are expressed by numeric
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literals, which consist of one or more decimal digits optionally preceeded by a minus sign (\isa{-}).  Examples are \isa{2}, \isa{-3} and
\isa{441223334678}.  Literals are available for the types of natural
numbers, integers, rationals, reals, etc.; they denote integer values of
arbitrary size.
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Literals look like constants, but they abbreviate 
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terms representing the number in a two's complement binary notation. 
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Isabelle performs arithmetic on literals by rewriting rather 
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than using the hardware arithmetic. In most cases arithmetic 
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is fast enough, even for numbers in the millions. The arithmetic operations 
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provided for literals include addition, subtraction, multiplication, 
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integer division and remainder.  Fractions of literals (expressed using
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division) are reduced to lowest terms.
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\begin{warn}\index{overloading!and arithmetic}
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The arithmetic operators are 
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overloaded, so you must be careful to ensure that each numeric 
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expression refers to a specific type, if necessary by inserting 
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type constraints.  Here is an example of what can go wrong:
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\par
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\begin{isabelle}
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\isacommand{lemma}\ "2\ *\ m\ =\ m\ +\ m"
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\end{isabelle}
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%
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Carefully observe how Isabelle displays the subgoal:
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\begin{isabelle}
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\ 1.\ (2::'a)\ *\ m\ =\ m\ +\ m
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\end{isabelle}
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The type \isa{'a} given for the literal \isa{2} warns us that no numeric
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type has been specified.  The problem is underspecified.  Given a type
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constraint such as \isa{nat}, \isa{int} or \isa{real}, it becomes trivial.
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\end{warn}
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\begin{warn}
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\index{recdef@\isacommand {recdef} (command)!and numeric literals}  
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Numeric literals are not constructors and therefore
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must not be used in patterns.  For example, this declaration is
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rejected:
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\begin{isabelle}
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\isacommand{recdef}\ h\ "\isacharbraceleft \isacharbraceright "\isanewline
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"h\ 3\ =\ 2"\isanewline
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"h\ i\ \ =\ i"
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\end{isabelle}
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You should use a conditional expression instead:
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\begin{isabelle}
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"h\ i\ =\ (if\ i\ =\ 3\ then\ 2\ else\ i)"
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\end{isabelle}
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\index{numeric literals|)}
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\end{warn}
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\subsection{The Type of Natural Numbers, {\tt\slshape nat}}
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\index{natural numbers|(}\index{*nat (type)|(}%
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This type requires no introduction: we have been using it from the
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beginning.  Hundreds of theorems about the natural numbers are
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proved in the theories \isa{Nat}, \isa{NatArith} and \isa{Divides}.  
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Basic properties of addition and multiplication are available through the
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axiomatic type class for semirings (\S\ref{sec:numeric-axclasses}).
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\subsubsection{Literals}
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\index{numeric literals!for type \protect\isa{nat}}%
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The notational options for the natural  numbers are confusing.  Recall that an
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overloaded constant can be defined independently for each type; the definition
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of \cdx{1} for type \isa{nat} is
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\begin{isabelle}
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1\ \isasymequiv\ Suc\ 0
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\rulename{One_nat_def}
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\end{isabelle}
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This is installed as a simplification rule, so the simplifier will replace
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every occurrence of \isa{1::nat} by \isa{Suc\ 0}.  Literals are obviously
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better than nested \isa{Suc}s at expressing large values.  But many theorems,
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including the rewrite rules for primitive recursive functions, can only be
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applied to terms of the form \isa{Suc\ $n$}.
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The following default  simplification rules replace
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small literals by zero and successor: 
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\begin{isabelle}
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2\ +\ n\ =\ Suc\ (Suc\ n)
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\rulename{add_2_eq_Suc}\isanewline
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n\ +\ 2\ =\ Suc\ (Suc\ n)
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\rulename{add_2_eq_Suc'}
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\end{isabelle}
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It is less easy to transform \isa{100} into \isa{Suc\ 99} (for example), and
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the simplifier will normally reverse this transformation.  Novices should
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express natural numbers using \isa{0} and \isa{Suc} only.
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\subsubsection{Division}
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\index{division!for type \protect\isa{nat}}%
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The infix operators \isa{div} and \isa{mod} are overloaded.
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Isabelle/HOL provides the basic facts about quotient and remainder
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on the natural numbers:
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\begin{isabelle}
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m\ mod\ n\ =\ (if\ m\ <\ n\ then\ m\ else\ (m\ -\ n)\ mod\ n)
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\rulename{mod_if}\isanewline
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m\ div\ n\ *\ n\ +\ m\ mod\ n\ =\ m%
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\rulenamedx{mod_div_equality}
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\end{isabelle}
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Many less obvious facts about quotient and remainder are also provided. 
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Here is a selection:
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\begin{isabelle}
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a\ *\ b\ div\ c\ =\ a\ *\ (b\ div\ c)\ +\ a\ *\ (b\ mod\ c)\ div\ c%
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\rulename{div_mult1_eq}\isanewline
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a\ *\ b\ mod\ c\ =\ a\ *\ (b\ mod\ c)\ mod\ c%
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\rulename{mod_mult1_eq}\isanewline
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a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%
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\rulename{div_mult2_eq}\isanewline
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a\ mod\ (b*c)\ =\ b * (a\ div\ b\ mod\ c)\ +\ a\ mod\ b%
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\rulename{mod_mult2_eq}\isanewline
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0\ <\ c\ \isasymLongrightarrow \ (c\ *\ a)\ div\ (c\ *\ b)\ =\ a\ div\ b%
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\rulename{div_mult_mult1}\isanewline
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(m\ mod\ n)\ *\ k\ =\ (m\ *\ k)\ mod\ (n\ *\ k)
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\rulenamedx{mod_mult_distrib}\isanewline
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m\ \isasymle \ n\ \isasymLongrightarrow \ m\ div\ k\ \isasymle \ n\ div\ k%
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\rulename{div_le_mono}
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\end{isabelle}
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Surprisingly few of these results depend upon the
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divisors' being nonzero.
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\index{division!by zero}%
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That is because division by
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zero yields zero:
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\begin{isabelle}
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a\ div\ 0\ =\ 0
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\rulename{DIVISION_BY_ZERO_DIV}\isanewline
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a\ mod\ 0\ =\ a%
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\rulename{DIVISION_BY_ZERO_MOD}
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\end{isabelle}
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In \isa{div_mult_mult1} above, one of
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the two divisors (namely~\isa{c}) must still be nonzero.
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The \textbf{divides} relation\index{divides relation}
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has the standard definition, which
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is overloaded over all numeric types: 
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\begin{isabelle}
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m\ dvd\ n\ \isasymequiv\ {\isasymexists}k.\ n\ =\ m\ *\ k
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\rulenamedx{dvd_def}
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\end{isabelle}
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%
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Section~\ref{sec:proving-euclid} discusses proofs involving this
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relation.  Here are some of the facts proved about it:
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\begin{isabelle}
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\isasymlbrakk m\ dvd\ n;\ n\ dvd\ m\isasymrbrakk \ \isasymLongrightarrow \ m\ =\ n%
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\rulenamedx{dvd_anti_sym}\isanewline
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\isasymlbrakk k\ dvd\ m;\ k\ dvd\ n\isasymrbrakk \ \isasymLongrightarrow \ k\ dvd\ (m\ +\ n)
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\rulenamedx{dvd_add}
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\end{isabelle}
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\subsubsection{Subtraction}
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There are no negative natural numbers, so \isa{m\ -\ n} equals zero unless 
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\isa{m} exceeds~\isa{n}. The following is one of the few facts
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about \isa{m\ -\ n} that is not subject to
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the condition \isa{n\ \isasymle \  m}. 
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\begin{isabelle}
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(m\ -\ n)\ *\ k\ =\ m\ *\ k\ -\ n\ *\ k%
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\rulenamedx{diff_mult_distrib}
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\end{isabelle}
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Natural number subtraction has few
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nice properties; often you should remove it by simplifying with this split
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rule.
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\begin{isabelle}
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P(a-b)\ =\ ((a<b\ \isasymlongrightarrow \ P\
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0)\ \isasymand \ (\isasymforall d.\ a\ =\ b+d\ \isasymlongrightarrow \ P\
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d))
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\rulename{nat_diff_split}
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\end{isabelle}
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For example, splitting helps to prove the following fact.
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\begin{isabelle}
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\isacommand{lemma}\ "(n\ -\ 2)\ *\ (n\ +\ 2)\ =\ n\ *\ n\ -\ (4::nat)"\isanewline
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\isacommand{apply}\ (simp\ split:\ nat_diff_split,\ clarify)\isanewline
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\ 1.\ \isasymAnd d.\ \isasymlbrakk n\ <\ 2;\ n\ *\ n\ =\ 4\ +\ d\isasymrbrakk \ \isasymLongrightarrow \ d\ =\ 0
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\end{isabelle}
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The result lies outside the scope of linear arithmetic, but
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 it is easily found
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if we explicitly split \isa{n<2} as \isa{n=0} or \isa{n=1}:
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\begin{isabelle}
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\isacommand{apply}\ (subgoal_tac\ "n=0\ |\ n=1",\ force,\ arith)\isanewline
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\isacommand{done}
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\end{isabelle}%%%%%%
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\index{natural numbers|)}\index{*nat (type)|)}
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\subsection{The Type of Integers, {\tt\slshape int}}
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\index{integers|(}\index{*int (type)|(}%
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Reasoning methods for the integers resemble those for the natural numbers, 
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but induction and
the constant \isa{Suc} are not available.  HOL provides many lemmas for
proving inequalities involving integer multiplication and division, similar
to those shown above for type~\isa{nat}. The laws of addition, subtraction
and multiplication are available through the axiomatic type class for rings
(\S\ref{sec:numeric-axclasses}).
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The \rmindex{absolute value} function \cdx{abs} is overloaded, and is 
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defined for all types that involve negative numbers, including the integers.
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The \isa{arith} method can prove facts about \isa{abs} automatically, 
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though as it does so by case analysis, the cost can be exponential.
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\begin{isabelle}
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\isacommand{lemma}\ "abs\ (x+y)\ \isasymle \ abs\ x\ +\ abs\ (y\ ::\ int)"\isanewline
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\isacommand{by}\ arith
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\end{isabelle}
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For division and remainder,\index{division!by negative numbers}
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the treatment of negative divisors follows
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mathematical practice: the sign of the remainder follows that
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of the divisor:
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\begin{isabelle}
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0\ <\ b\ \isasymLongrightarrow \ 0\ \isasymle \ a\ mod\ b%
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\rulename{pos_mod_sign}\isanewline
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0\ <\ b\ \isasymLongrightarrow \ a\ mod\ b\ <\ b%
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\rulename{pos_mod_bound}\isanewline
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b\ <\ 0\ \isasymLongrightarrow \ a\ mod\ b\ \isasymle \ 0
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\rulename{neg_mod_sign}\isanewline
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b\ <\ 0\ \isasymLongrightarrow \ b\ <\ a\ mod\ b%
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\rulename{neg_mod_bound}
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\end{isabelle}
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ML treats negative divisors in the same way, but most computer hardware
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treats signed operands using the same rules as for multiplication.
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Many facts about quotients and remainders are provided:
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\begin{isabelle}
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(a\ +\ b)\ div\ c\ =\isanewline
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a\ div\ c\ +\ b\ div\ c\ +\ (a\ mod\ c\ +\ b\ mod\ c)\ div\ c%
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\rulename{zdiv_zadd1_eq}
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\par\smallskip
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(a\ +\ b)\ mod\ c\ =\ (a\ mod\ c\ +\ b\ mod\ c)\ mod\ c%
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\rulename{zmod_zadd1_eq}
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\end{isabelle}
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\begin{isabelle}
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(a\ *\ b)\ div\ c\ =\ a\ *\ (b\ div\ c)\ +\ a\ *\ (b\ mod\ c)\ div\ c%
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\rulename{zdiv_zmult1_eq}\isanewline
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(a\ *\ b)\ mod\ c\ =\ a\ *\ (b\ mod\ c)\ mod\ c%
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\rulename{zmod_zmult1_eq}
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\end{isabelle}
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\begin{isabelle}
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0\ <\ c\ \isasymLongrightarrow \ a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%
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\rulename{zdiv_zmult2_eq}\isanewline
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0\ <\ c\ \isasymLongrightarrow \ a\ mod\ (b*c)\ =\ b*(a\ div\ b\ mod\
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c)\ +\ a\ mod\ b%
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\rulename{zmod_zmult2_eq}
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\end{isabelle}
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The last two differ from their natural number analogues by requiring
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\isa{c} to be positive.  Since division by zero yields zero, we could allow
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\isa{c} to be zero.  However, \isa{c} cannot be negative: a counterexample
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is
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$\isa{a} = 7$, $\isa{b} = 2$ and $\isa{c} = -3$, when the left-hand side of
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\isa{zdiv_zmult2_eq} is $-2$ while the right-hand side is~$-1$.
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The prefix~\isa{z} in many theorem names recalls the use of $\mathbb{Z}$ to
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denote the set of integers.%
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\index{integers|)}\index{*int (type)|)}
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Induction is less important for integers than it is for the natural numbers, but it can be valuable if the range of integers has a lower or upper bound.  There are four rules for integer induction, corresponding to the possible relations of the bound ($\geq$, $>$, $\leq$ and $<$):
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\begin{isabelle}
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\isasymlbrakk k\ \isasymle \ i;\ P\ k;\ \isasymAnd i.\ \isasymlbrakk k\ \isasymle \ i;\ P\ i\isasymrbrakk \ \isasymLongrightarrow \ P(i+1)\isasymrbrakk \ \isasymLongrightarrow \ P\ i%
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\rulename{int_ge_induct}\isanewline
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\isasymlbrakk k\ <\ i;\ P(k+1);\ \isasymAnd i.\ \isasymlbrakk k\ <\ i;\ P\ i\isasymrbrakk \ \isasymLongrightarrow \ P(i+1)\isasymrbrakk \ \isasymLongrightarrow \ P\ i%
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\rulename{int_gr_induct}\isanewline
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\isasymlbrakk i\ \isasymle \ k;\ P\ k;\ \isasymAnd i.\ \isasymlbrakk i\ \isasymle \ k;\ P\ i\isasymrbrakk \ \isasymLongrightarrow \ P(i-1)\isasymrbrakk \ \isasymLongrightarrow \ P\ i%
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\rulename{int_le_induct}\isanewline
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\isasymlbrakk i\ <\ k;\ P(k-1);\ \isasymAnd i.\ \isasymlbrakk i\ <\ k;\ P\ i\isasymrbrakk \ \isasymLongrightarrow \ P(i-1)\isasymrbrakk \ \isasymLongrightarrow \ P\ i%
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\rulename{int_less_induct}
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\end{isabelle}
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   309
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   310
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   311
\subsection{The Types of Rational, Real and Complex Numbers}
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\label{sec:real}
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\index{rational numbers|(}\index{*rat (type)|(}%
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   315
\index{real numbers|(}\index{*real (type)|(}%
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   316
\index{complex numbers|(}\index{*complex (type)|(}%
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   317
These types provide true division, the overloaded operator \isa{/}, 
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which differs from the operator \isa{div} of the 
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natural numbers and integers. The rationals and reals are 
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\textbf{dense}: between every two distinct numbers lies another.
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This property follows from the division laws, since if $x\not=y$ then $(x+y)/2$ lies between them:
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\begin{isabelle}
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a\ <\ b\ \isasymLongrightarrow \ \isasymexists r.\ a\ <\ r\ \isasymand \ r\ <\ b%
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\rulename{dense}
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\end{isabelle}
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The real numbers are, moreover, \textbf{complete}: every set of reals that
is bounded above has a least upper bound.  Completeness distinguishes the
reals from the rationals, for which the set $\{x\mid x^2<2\}$ has no least
upper bound.  (It could only be $\surd2$, which is irrational. The
formalization of completeness, which is complicated, 
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can be found in theory \texttt{RComplete} of directory
\texttt{Real}.
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   329
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   330
Numeric literals\index{numeric literals!for type \protect\isa{real}}
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for type \isa{real} have the same syntax as those for type
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\isa{int} and only express integral values.  Fractions expressed
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using the division operator are automatically simplified to lowest terms:
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   334
\begin{isabelle}
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\ 1.\ P\ ((3\ /\ 4)\ *\ (8\ /\ 15))\isanewline
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   336
\isacommand{apply} simp\isanewline
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   337
\ 1.\ P\ (2\ /\ 5)
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   338
\end{isabelle}
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   339
Exponentiation can express floating-point values such as
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   340
\isa{2 * 10\isacharcircum6}, but at present no special simplification
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   341
is performed.
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   342
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   343
\begin{warn}
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Type \isa{real} is only available in the logic HOL-Complex, which is
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   345
HOL extended with a definitional development of the real and complex
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   346
numbers.  Base your theory upon theory \thydx{Complex_Main}, not the
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usual \isa{Main}, and set the Proof General menu item \pgmenu{Isabelle} $>$
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\pgmenu{Logics} $>$ \pgmenu{HOL-Complex}.%
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   349
\index{real numbers|)}\index{*real (type)|)}
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   350
\end{warn}
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   351
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   352
Also available in HOL-Complex is the
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theory \isa{Hyperreal}, which define the type \tydx{hypreal} of 
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   354
\rmindex{non-standard reals}.  These
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   355
\textbf{hyperreals} include infinitesimals, which represent infinitely
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   356
small and infinitely large quantities; they facilitate proofs
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   357
about limits, differentiation and integration~\cite{fleuriot-jcm}.  The
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   358
development defines an infinitely large number, \isa{omega} and an
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   359
infinitely small positive number, \isa{epsilon}.  The 
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   360
relation $x\approx y$ means ``$x$ is infinitely close to~$y$.''
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   361
Theory \isa{Hyperreal} also defines transcendental functions such as sine,
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   362
cosine, exponential and logarithm --- even the versions for type
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   363
\isa{real}, because they are defined using nonstandard limits.%
paulson@14400
   364
\index{rational numbers|)}\index{*rat (type)|)}%
paulson@14400
   365
\index{real numbers|)}\index{*real (type)|)}%
paulson@14400
   366
\index{complex numbers|)}\index{*complex (type)|)}
paulson@14400
   367
paulson@14400
   368
paulson@14400
   369
\subsection{The Numeric Type Classes}\label{sec:numeric-axclasses}
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   370
paulson@14400
   371
Isabelle/HOL organises its numeric theories using axiomatic type classes.
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   372
Hundreds of basic properties are proved in the theory \isa{Ring_and_Field}.
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   373
These lemmas are available (as simprules if they were declared as such)
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   374
for all numeric types satisfying the necessary axioms. The theory defines
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   375
the following type classes:
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   376
\begin{itemize}
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   377
\item 
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   378
\tcdx{semiring} and \tcdx{ordered_semiring}: a \emph{semiring}
provides the operators \isa{+} and~\isa{*}, which are commutative and
associative, with the usual distributive law and with \isa{0} and~\isa{1}
as their respective identities. An \emph{ordered semiring} is also linearly
ordered, with addition and multiplication respecting the ordering. Type \isa{nat} is an ordered semiring.
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   379
\item 
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   380
\tcdx{ring} and \tcdx{ordered_ring}: a \emph{ring} extends a semiring
with unary minus (the additive inverse) and subtraction (both
denoted~\isa{-}). An \emph{ordered ring} includes the absolute value
function, \cdx{abs}. Type \isa{int} is an ordered ring.
paulson@14400
   381
\item 
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   382
\tcdx{field} and \tcdx{ordered_field}: a field extends a ring with the
multiplicative inverse (called simply \cdx{inverse} and division~(\isa{/}).
An ordered field is based on an ordered ring. Type \isa{complex} is a field, while type \isa{real} is an ordered field.
paulson@14400
   383
\item 
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   384
\tcdx{division_by_zero} includes all types where \isa{inverse 0 = 0}
paulson@14400
   385
and \isa{a / 0 = 0}. These include all of Isabelle's standard numeric types.
paulson@14400
   386
However, the basic properties of fields are derived without assuming
paulson@14400
   387
division by zero.
\end{itemize}
paulson@14400
   388
paulson@14400
   389
Theory \thydx{Ring_and_Field} proves over 250 lemmas, each of which
holds for all types in the corresponding type class. In most
cases, it is obvious whether a property is valid for a particular type. All
abstract properties involving subtraction require a ring, and therefore do
not hold for type \isa{nat}, although we have theorems such as
\isa{diff_mult_distrib} proved specifically about subtraction on
type~\isa{nat}. All abstract properties involving division require a field.
Obviously, all properties involving orderings required an ordered
structure.
paulson@14400
   390
paulson@14400
   391
The following two theorems are less obvious. Although they
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   392
mention no ordering, they require an ordered ring. However, if we have a 
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   393
field, then an ordering is no longer required.
paulson@14400
   394
\begin{isabelle}
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   395
(a\ *\ b\ =\ (0::'a))\ =\ (a\ =\ (0::'a)\ \isasymor \ b\ =\ (0::'a))
paulson@14400
   396
\rulename{mult_eq_0_iff}\isanewline
paulson@14400
   397
(a\ *\ c\ =\ b\ *\ c)\ =\ (c\ =\ (0::'a)\ \isasymor \ a\ =\ b)
paulson@14400
   398
\rulename{mult_cancel_right}
paulson@14400
   399
\end{isabelle}
paulson@14400
   400
Theorems \isa{field_mult_eq_0_iff} and \isa{field_mult_cancel_right}
nipkow@16412
   401
express the same properties, only for fields.
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   402
\begin{pgnote}
nipkow@16523
   403
Setting the flag \pgmenu{Isabelle} $>$ \pgmenu{Settings} $>$
nipkow@16523
   404
\pgmenu{Show Sorts} will display the type classes of all type variables.
nipkow@16412
   405
\end{pgnote}
nipkow@16412
   406
\noindent
nipkow@16412
   407
Here is how the theorem \isa{field_mult_cancel_right} appears with the flag set.
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   408
\begin{isabelle}
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   409
((a::'a::field)\ *\ (c::'a::field)\ =\ (b::'a::field)\ *\ c)\ =\isanewline
paulson@14400
   410
(c\ =\ (0::'a::field)\ \isasymor \ a\ =\ b)
paulson@14400
   411
\end{isabelle}
paulson@14400
   412
paulson@14400
   413
paulson@14400
   414
\subsubsection{Simplifying with the AC-Laws}
paulson@14400
   415
Suppose that two expressions are equal, differing only in 
paulson@14400
   416
associativity and commutativity of addition.  Simplifying with the
paulson@14400
   417
following equations sorts the terms and groups them to the right, making
paulson@14400
   418
the two expressions identical.
paulson@14400
   419
\begin{isabelle}
paulson@14400
   420
a\ +\ b\ +\ c\ =\ a\ +\ (b\ +\ c)
paulson@14400
   421
\rulenamedx{add_assoc}\isanewline
paulson@14400
   422
a\ +\ b\ =\ b\ +\ a%
paulson@14400
   423
\rulenamedx{add_commute}\isanewline
paulson@14400
   424
a\ +\ (b\ +\ c)\ =\ b\ +\ (a\ +\ c)
paulson@14400
   425
\rulename{add_left_commute}
paulson@14400
   426
\end{isabelle}
paulson@14400
   427
The name \isa{add_ac}\index{*add_ac (theorems)} 
paulson@14400
   428
refers to the list of all three theorems; similarly
paulson@14400
   429
there is \isa{mult_ac}.\index{*mult_ac (theorems)} 
paulson@14400
   430
They are all proved for semirings and therefore hold for all numeric types.
paulson@14400
   431
paulson@14400
   432
Here is an example of the sorting effect.  Start
paulson@14400
   433
with this goal, which involves type \isa{nat}.
paulson@14400
   434
\begin{isabelle}
paulson@14400
   435
\ 1.\ Suc\ (i\ +\ j\ *\ l\ *\ k\ +\ m\ *\ n)\ =\
paulson@14400
   436
f\ (n\ *\ m\ +\ i\ +\ k\ *\ j\ *\ l)
paulson@14400
   437
\end{isabelle}
paulson@14400
   438
%
paulson@14400
   439
Simplify using  \isa{add_ac} and \isa{mult_ac}.
paulson@14400
   440
\begin{isabelle}
paulson@14400
   441
\isacommand{apply}\ (simp\ add:\ add_ac\ mult_ac)
paulson@14400
   442
\end{isabelle}
paulson@14400
   443
%
paulson@14400
   444
Here is the resulting subgoal.
paulson@14400
   445
\begin{isabelle}
paulson@14400
   446
\ 1.\ Suc\ (i\ +\ (m\ *\ n\ +\ j\ *\ (k\ *\ l)))\
paulson@14400
   447
=\ f\ (i\ +\ (m\ *\ n\ +\ j\ *\ (k\ *\ l)))%
paulson@14400
   448
\end{isabelle}
paulson@14400
   449
paulson@14400
   450
paulson@14400
   451
\subsubsection{Division Laws for Fields}
paulson@14400
   452
paulson@10777
   453
Here is a selection of rules about the division operator.  The following
paulson@10777
   454
are installed as default simplification rules in order to express
paulson@10777
   455
combinations of products and quotients as rational expressions:
paulson@10777
   456
\begin{isabelle}
paulson@14288
   457
a\ *\ (b\ /\ c)\ =\ a\ *\ b\ /\ c
paulson@14288
   458
\rulename{times_divide_eq_right}\isanewline
paulson@14288
   459
b\ /\ c\ *\ a\ =\ b\ *\ a\ /\ c
paulson@14288
   460
\rulename{times_divide_eq_left}\isanewline
paulson@14288
   461
a\ /\ (b\ /\ c)\ =\ a\ *\ c\ /\ b
paulson@14288
   462
\rulename{divide_divide_eq_right}\isanewline
paulson@14288
   463
a\ /\ b\ /\ c\ =\ a\ /\ (b\ *\ c)
paulson@14288
   464
\rulename{divide_divide_eq_left}
paulson@10777
   465
\end{isabelle}
paulson@10777
   466
paulson@10777
   467
Signs are extracted from quotients in the hope that complementary terms can
paulson@10777
   468
then be cancelled:
paulson@10777
   469
\begin{isabelle}
paulson@14295
   470
-\ (a\ /\ b)\ =\ -\ a\ /\ b
paulson@14295
   471
\rulename{minus_divide_left}\isanewline
paulson@14295
   472
-\ (a\ /\ b)\ =\ a\ /\ -\ b
paulson@14295
   473
\rulename{minus_divide_right}
paulson@10777
   474
\end{isabelle}
paulson@10777
   475
paulson@10777
   476
The following distributive law is available, but it is not installed as a
paulson@10777
   477
simplification rule.
paulson@10777
   478
\begin{isabelle}
paulson@14295
   479
(a\ +\ b)\ /\ c\ =\ a\ /\ c\ +\ b\ /\ c%
paulson@14295
   480
\rulename{add_divide_distrib}
paulson@10777
   481
\end{isabelle}
paulson@10777
   482
paulson@14400
   483
paulson@14400
   484
\subsubsection{Absolute Value}
paulson@10594
   485
paulson@14400
   486
The \rmindex{absolute value} function \cdx{abs} is available for all 
paulson@14400
   487
ordered rings, including types \isa{int}, \isa{rat} and \isa{real}.
paulson@14400
   488
It satisfies many properties,
paulson@14400
   489
such as the following:
paulson@10777
   490
\begin{isabelle}
paulson@14400
   491
\isasymbar x\ *\ y\isasymbar \ =\ \isasymbar x\isasymbar \ *\ \isasymbar y\isasymbar 
paulson@14400
   492
\rulename{abs_mult}\isanewline
paulson@14400
   493
(\isasymbar a\isasymbar \ \isasymle \ b)\ =\ (a\ \isasymle \ b\ \isasymand \ -\ a\ \isasymle \ b)
paulson@14400
   494
\rulename{abs_le_iff}\isanewline
paulson@14400
   495
\isasymbar a\ +\ b\isasymbar \ \isasymle \ \isasymbar a\isasymbar \ +\ \isasymbar b\isasymbar 
paulson@14400
   496
\rulename{abs_triangle_ineq}
paulson@10777
   497
\end{isabelle}
paulson@10777
   498
paulson@14400
   499
\begin{warn}
paulson@14400
   500
The absolute value bars shown above cannot be typed on a keyboard.  They
paulson@14400
   501
can be entered using the X-symbol package.  In \textsc{ascii}, type \isa{abs x} to
paulson@14400
   502
get \isa{\isasymbar x\isasymbar}.
paulson@14400
   503
\end{warn}
paulson@11174
   504
paulson@11174
   505
paulson@14400
   506
\subsubsection{Raising to a Power}
paulson@10777
   507
paulson@14400
   508
Another type class, \tcdx{ringppower}, specifies rings that also have 
paulson@14400
   509
exponentation to a natural number power, defined using the obvious primitive
paulson@14400
   510
recursion. Theory \thydx{Power} proves various theorems, such as the 
paulson@14400
   511
following.
paulson@14400
   512
\begin{isabelle}
paulson@14400
   513
a\ \isacharcircum \ (m\ +\ n)\ =\ a\ \isacharcircum \ m\ *\ a\ \isacharcircum \ n%
paulson@14400
   514
\rulename{power_add}\isanewline
paulson@14400
   515
a\ \isacharcircum \ (m\ *\ n)\ =\ (a\ \isacharcircum \ m)\ \isacharcircum \ n%
paulson@14400
   516
\rulename{power_mult}\isanewline
paulson@14400
   517
\isasymbar a\ \isacharcircum \ n\isasymbar \ =\ \isasymbar a\isasymbar \ \isacharcircum \ n%
paulson@14400
   518
\rulename{power_abs}
paulson@14400
   519
\end{isabelle}%%%%%%%%%%%%%%%%%%%%%%%%%
nipkow@13996
   520
\index{numbers|)}