Clarification: free variables allowed in interpreted locale instances.
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\begin{isamarkuptext}%
\noindent
In particular, there are \isa{case}-expressions, for example
\begin{isabelle}%
\ \ \ \ \ case\ n\ of\ {\isadigit{0}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isadigit{0}}\ {\isaliteral{7C}{\isacharbar}}\ Suc\ m\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ m%
\end{isabelle}
primitive recursion, for example%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{primrec}\isamarkupfalse%
\ sum\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
{\isaliteral{22}{\isachardoublequoteopen}}sum\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
{\isaliteral{22}{\isachardoublequoteopen}}sum\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ n\ {\isaliteral{2B}{\isacharplus}}\ sum\ n{\isaliteral{22}{\isachardoublequoteclose}}%
\begin{isamarkuptext}%
\noindent
and induction, for example%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
\ {\isaliteral{22}{\isachardoublequoteopen}}sum\ n\ {\isaliteral{2B}{\isacharplus}}\ sum\ n\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
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\isadelimproof
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\endisadelimproof
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\isatagproof
\isacommand{apply}\isamarkupfalse%
{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ n{\isaliteral{29}{\isacharparenright}}\isanewline
\isacommand{apply}\isamarkupfalse%
{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}\isanewline
\isacommand{done}\isamarkupfalse%
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\endisatagproof
{\isafoldproof}%
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\isadelimproof
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\endisadelimproof
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\begin{isamarkuptext}%
\newcommand{\mystar}{*%
}
\index{arithmetic operations!for \protect\isa{nat}}%
The arithmetic operations \isadxboldpos{+}{$HOL2arithfun},
\isadxboldpos{-}{$HOL2arithfun}, \isadxboldpos{\mystar}{$HOL2arithfun},
\sdx{div}, \sdx{mod}, \cdx{min} and
\cdx{max} are predefined, as are the relations
\isadxboldpos{\isasymle}{$HOL2arithrel} and
\isadxboldpos{<}{$HOL2arithrel}. As usual, \isa{m\ {\isaliteral{2D}{\isacharminus}}\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}} if
\isa{m\ {\isaliteral{3C}{\isacharless}}\ n}. There is even a least number operation
\sdx{LEAST}\@. For example, \isa{{\isaliteral{28}{\isacharparenleft}}LEAST\ n{\isaliteral{2E}{\isachardot}}\ {\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isadigit{0}}}.
\begin{warn}\index{overloading}
The constants \cdx{0} and \cdx{1} and the operations
\isadxboldpos{+}{$HOL2arithfun}, \isadxboldpos{-}{$HOL2arithfun},
\isadxboldpos{\mystar}{$HOL2arithfun}, \cdx{min},
\cdx{max}, \isadxboldpos{\isasymle}{$HOL2arithrel} and
\isadxboldpos{<}{$HOL2arithrel} are overloaded: they are available
not just for natural numbers but for other types as well.
For example, given the goal \isa{x\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ x}, there is nothing to indicate
that you are talking about natural numbers. Hence Isabelle can only infer
that \isa{x} is of some arbitrary type where \isa{{\isadigit{0}}} and \isa{{\isaliteral{2B}{\isacharplus}}} are
declared. As a consequence, you will be unable to prove the
goal. To alert you to such pitfalls, Isabelle flags numerals without a
fixed type in its output: \isa{x\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ x}. (In the absence of a numeral,
it may take you some time to realize what has happened if \pgmenu{Show
Types} is not set). In this particular example, you need to include
an explicit type constraint, for example \isa{x{\isaliteral{2B}{\isacharplus}}{\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}}. If there
is enough contextual information this may not be necessary: \isa{Suc\ x\ {\isaliteral{3D}{\isacharequal}}\ x} automatically implies \isa{x{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat} because \isa{Suc} is not
overloaded.
For details on overloading see \S\ref{sec:overloading}.
Table~\ref{tab:overloading} in the appendix shows the most important
overloaded operations.
\end{warn}
\begin{warn}
The symbols \isadxboldpos{>}{$HOL2arithrel} and
\isadxboldpos{\isasymge}{$HOL2arithrel} are merely syntax: \isa{x\ {\isaliteral{3E}{\isachargreater}}\ y}
stands for \isa{y\ {\isaliteral{3C}{\isacharless}}\ x} and similary for \isa{{\isaliteral{5C3C67653E}{\isasymge}}} and
\isa{{\isaliteral{5C3C6C653E}{\isasymle}}}.
\end{warn}
\begin{warn}
Constant \isa{{\isadigit{1}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat} is defined to equal \isa{Suc\ {\isadigit{0}}}. This definition
(see \S\ref{sec:ConstDefinitions}) is unfolded automatically by some
tactics (like \isa{auto}, \isa{simp} and \isa{arith}) but not by
others (especially the single step tactics in Chapter~\ref{chap:rules}).
If you need the full set of numerals, see~\S\ref{sec:numerals}.
\emph{Novices are advised to stick to \isa{{\isadigit{0}}} and \isa{Suc}.}
\end{warn}
Both \isa{auto} and \isa{simp}
(a method introduced below, \S\ref{sec:Simplification}) prove
simple arithmetic goals automatically:%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ m\ {\isaliteral{3C}{\isacharless}}\ n{\isaliteral{3B}{\isacharsemicolon}}\ m\ {\isaliteral{3C}{\isacharless}}\ n\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ m\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{22}{\isachardoublequoteclose}}%
\isadelimproof
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\endisadelimproof
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\isatagproof
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\endisatagproof
{\isafoldproof}%
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\begin{isamarkuptext}%
\noindent
For efficiency's sake, this built-in prover ignores quantified formulae,
many logical connectives, and all arithmetic operations apart from addition.
In consequence, \isa{auto} and \isa{simp} cannot prove this slightly more complex goal:%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
\ {\isaliteral{22}{\isachardoublequoteopen}}m\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isaliteral{28}{\isacharparenleft}}n{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ m\ {\isaliteral{3C}{\isacharless}}\ n\ {\isaliteral{5C3C6F723E}{\isasymor}}\ n\ {\isaliteral{3C}{\isacharless}}\ m{\isaliteral{22}{\isachardoublequoteclose}}%
\isadelimproof
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\endisadelimproof
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\isatagproof
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\endisatagproof
{\isafoldproof}%
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\isadelimproof
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\begin{isamarkuptext}%
\noindent The method \methdx{arith} is more general. It attempts to
prove the first subgoal provided it is a \textbf{linear arithmetic} formula.
Such formulas may involve the usual logical connectives (\isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}},
\isa{{\isaliteral{5C3C616E643E}{\isasymand}}}, \isa{{\isaliteral{5C3C6F723E}{\isasymor}}}, \isa{{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}}, \isa{{\isaliteral{3D}{\isacharequal}}},
\isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}}, \isa{{\isaliteral{5C3C6578697374733E}{\isasymexists}}}), the relations \isa{{\isaliteral{3D}{\isacharequal}}},
\isa{{\isaliteral{5C3C6C653E}{\isasymle}}} and \isa{{\isaliteral{3C}{\isacharless}}}, and the operations \isa{{\isaliteral{2B}{\isacharplus}}}, \isa{{\isaliteral{2D}{\isacharminus}}},
\isa{min} and \isa{max}. For example,%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
\ {\isaliteral{22}{\isachardoublequoteopen}}min\ i\ {\isaliteral{28}{\isacharparenleft}}max\ j\ {\isaliteral{28}{\isacharparenleft}}k{\isaliteral{2A}{\isacharasterisk}}k{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ max\ {\isaliteral{28}{\isacharparenleft}}min\ {\isaliteral{28}{\isacharparenleft}}k{\isaliteral{2A}{\isacharasterisk}}k{\isaliteral{29}{\isacharparenright}}\ i{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}min\ i\ {\isaliteral{28}{\isacharparenleft}}j{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
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\isadelimproof
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\endisadelimproof
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\isatagproof
\isacommand{apply}\isamarkupfalse%
{\isaliteral{28}{\isacharparenleft}}arith{\isaliteral{29}{\isacharparenright}}%
\endisatagproof
{\isafoldproof}%
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\isadelimproof
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\begin{isamarkuptext}%
\noindent
succeeds because \isa{k\ {\isaliteral{2A}{\isacharasterisk}}\ k} can be treated as atomic. In contrast,%
\end{isamarkuptext}%
\isamarkuptrue%
\isacommand{lemma}\isamarkupfalse%
\ {\isaliteral{22}{\isachardoublequoteopen}}n{\isaliteral{2A}{\isacharasterisk}}n\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ n{\isaliteral{3D}{\isacharequal}}{\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}%
\isadelimproof
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\endisadelimproof
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\isatagproof
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\endisatagproof
{\isafoldproof}%
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\isadelimproof
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\endisadelimproof
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\begin{isamarkuptext}%
\noindent
is not proved by \isa{arith} because the proof relies
on properties of multiplication. Only multiplication by numerals (which is
the same as iterated addition) is taken into account.
\begin{warn} The running time of \isa{arith} is exponential in the number
of occurrences of \ttindexboldpos{-}{$HOL2arithfun}, \cdx{min} and
\cdx{max} because they are first eliminated by case distinctions.
If \isa{k} is a numeral, \sdx{div}~\isa{k}, \sdx{mod}~\isa{k} and
\isa{k}~\sdx{dvd} are also supported, where the former two are eliminated
by case distinctions, again blowing up the running time.
If the formula involves quantifiers, \isa{arith} may take
super-exponential time and space.
\end{warn}%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimtheory
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\isatagtheory
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\endisatagtheory
{\isafoldtheory}%
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\end{isabellebody}%
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