(*<*)theory Logic imports Main begin(*>*)

section{*Logic \label{sec:Logic}*}

subsection{*Propositional logic*}

subsubsection{*Introduction rules*}

text{* We start with a really trivial toy proof to introduce the basic

features of structured proofs. *}

lemma "A \<longrightarrow> A"

proof (rule impI)

  assume a: "A"

  show "A" by(rule a)

qed

text{*\noindent

The operational reading: the \isakeyword{assume}-\isakeyword{show}

block proves @{prop"A \<Longrightarrow> A"} (@{term a} is a degenerate rule (no

assumptions) that proves @{term A} outright), which rule

@{thm[source]impI} (@{thm impI}) turns into the desired @{prop"A \<longrightarrow>

A"}.  However, this text is much too detailed for comfort. Therefore

Isar implements the following principle: \begin{quote}\em Command

\isakeyword{proof} automatically tries to select an introduction rule

based on the goal and a predefined list of rules.  \end{quote} Here

@{thm[source]impI} is applied automatically: *}

lemma "A \<longrightarrow> A"

proof

  assume a: A

  show A by(rule a)

qed

text{*\noindent Single-identifier formulae such as @{term A} need not

be enclosed in double quotes. However, we will continue to do so for

uniformity.

Trivial proofs, in particular those by assumption, should be trivial

to perform. Proof ``.'' does just that (and a bit more). Thus

naming of assumptions is often superfluous: *}

lemma "A \<longrightarrow> A"

proof

  assume "A"

  show "A" .

qed

text{* To hide proofs by assumption further, \isakeyword{by}@{text"(method)"}

first applies @{text method} and then tries to solve all remaining subgoals

by assumption: *}

lemma "A \<longrightarrow> A \<and> A"

proof

  assume "A"

  show "A \<and> A" by(rule conjI)

qed

text{*\noindent Rule @{thm[source]conjI} is of course @{thm conjI}.

A drawback of implicit proofs by assumption is that it

is no longer obvious where an assumption is used.

Proofs of the form \isakeyword{by}@{text"(rule"}~\emph{name}@{text")"}

can be abbreviated to ``..''  if \emph{name} refers to one of the

predefined introduction rules (or elimination rules, see below): *}

lemma "A \<longrightarrow> A \<and> A"

proof

  assume "A"

  show "A \<and> A" ..

qed

text{*\noindent

This is what happens: first the matching introduction rule @{thm[source]conjI}

is applied (first ``.''), then the two subgoals are solved by assumption

(second ``.''). *}

subsubsection{*Elimination rules*}

text{*A typical elimination rule is @{thm[source]conjE}, $\land$-elimination:

@{thm[display,indent=5]conjE}  In the following proof it is applied

by hand, after its first (\emph{major}) premise has been eliminated via

@{text"[OF AB]"}: *}

lemma "A \<and> B \<longrightarrow> B \<and> A"

proof

  assume AB: "A \<and> B"

  show "B \<and> A"

  proof (rule conjE[OF AB])  -- {*@{text"conjE[OF AB]"}: @{thm conjE[OF AB]} *}

    assume "A" "B"

    show ?thesis ..

  qed

qed

text{*\noindent Note that the term @{text"?thesis"} always stands for the

``current goal'', i.e.\ the enclosing \isakeyword{show} (or

\isakeyword{have}) statement.

This is too much proof text. Elimination rules should be selected

automatically based on their major premise, the formula or rather connective

to be eliminated. In Isar they are triggered by facts being fed

\emph{into} a proof. Syntax:

\begin{center}

\isakeyword{from} \emph{fact} \isakeyword{show} \emph{proposition} \emph{proof}

\end{center}

where \emph{fact} stands for the name of a previously proved

proposition, e.g.\ an assumption, an intermediate result or some global

theorem, which may also be modified with @{text OF} etc.

The \emph{fact} is ``piped'' into the \emph{proof}, which can deal with it

how it chooses. If the \emph{proof} starts with a plain \isakeyword{proof},

an elimination rule (from a predefined list) is applied

whose first premise is solved by the \emph{fact}. Thus the proof above

is equivalent to the following one: *}

lemma "A \<and> B \<longrightarrow> B \<and> A"

proof

  assume AB: "A \<and> B"

  from AB show "B \<and> A"

  proof

    assume "A" "B"

    show ?thesis ..

  qed

qed

text{* Now we come to a second important principle:

\begin{quote}\em

Try to arrange the sequence of propositions in a UNIX-like pipe,

such that the proof of each proposition builds on the previous proposition.

\end{quote}

The previous proposition can be referred to via the fact @{text this}.

This greatly reduces the need for explicit naming of propositions:

*}

lemma "A \<and> B \<longrightarrow> B \<and> A"

proof

  assume "A \<and> B"

  from this show "B \<and> A"

  proof

    assume "A" "B"

    show ?thesis ..

  qed

qed

text{*\noindent Because of the frequency of \isakeyword{from}~@{text

this}, Isar provides two abbreviations:

\begin{center}

\begin{tabular}{r@ {\quad=\quad}l}

\isakeyword{then} & \isakeyword{from} @{text this} \\

\isakeyword{thus} & \isakeyword{then} \isakeyword{show}

\end{tabular}

\end{center}

Here is an alternative proof that operates purely by forward reasoning: *}

lemma "A \<and> B \<longrightarrow> B \<and> A"

proof

  assume ab: "A \<and> B"

  from ab have a: "A" ..

  from ab have b: "B" ..

  from b a show "B \<and> A" ..

qed

text{*\noindent It is worth examining this text in detail because it

exhibits a number of new concepts.  For a start, it is the first time

we have proved intermediate propositions (\isakeyword{have}) on the

way to the final \isakeyword{show}. This is the norm in nontrivial

proofs where one cannot bridge the gap between the assumptions and the

conclusion in one step. To understand how the proof works we need to

explain more Isar details.

Method @{text rule} can be given a list of rules, in which case

@{text"(rule"}~\textit{rules}@{text")"} applies the first matching

rule in the list \textit{rules}. Command \isakeyword{from} can be

followed by any number of facts.  Given \isakeyword{from}~@{text

f}$_1$~\dots~@{text f}$_n$, the proof step

@{text"(rule"}~\textit{rules}@{text")"} following a \isakeyword{have}

or \isakeyword{show} searches \textit{rules} for a rule whose first

$n$ premises can be proved by @{text f}$_1$~\dots~@{text f}$_n$ in the

given order. Finally one needs to know that ``..'' is short for

@{text"by(rule"}~\textit{elim-rules intro-rules}@{text")"} (or

@{text"by(rule"}~\textit{intro-rules}@{text")"} if there are no facts

fed into the proof), i.e.\ elimination rules are tried before

introduction rules.

Thus in the above proof both \isakeyword{have}s are proved via

@{thm[source]conjE} triggered by \isakeyword{from}~@{text ab} whereas

in the \isakeyword{show} step no elimination rule is applicable and

the proof succeeds with @{thm[source]conjI}. The latter would fail had

we written \isakeyword{from}~@{text"a b"} instead of

\isakeyword{from}~@{text"b a"}.

Proofs starting with a plain @{text proof} behave the same because the

latter is short for @{text"proof (rule"}~\textit{elim-rules

intro-rules}@{text")"} (or @{text"proof

(rule"}~\textit{intro-rules}@{text")"} if there are no facts fed into

the proof). *}

subsection{*More constructs*}

text{* In the previous proof of @{prop"A \<and> B \<longrightarrow> B \<and> A"} we needed to feed

more than one fact into a proof step, a frequent situation. Then the

UNIX-pipe model appears to break down and we need to name the different

facts to refer to them. But this can be avoided:

*}

lemma "A \<and> B \<longrightarrow> B \<and> A"

proof

  assume ab: "A \<and> B"

  from ab have "B" ..

  moreover

  from ab have "A" ..

  ultimately show "B \<and> A" ..

qed

text{*\noindent You can combine any number of facts @{term A1} \dots\ @{term

An} into a sequence by separating their proofs with

\isakeyword{moreover}. After the final fact, \isakeyword{ultimately} stands

for \isakeyword{from}~@{term A1}~\dots~@{term An}.  This avoids having to

introduce names for all of the sequence elements.  *}

text{* Although we have only seen a few introduction and elimination rules so

far, Isar's predefined rules include all the usual natural deduction

rules. We conclude our exposition of propositional logic with an extended

example --- which rules are used implicitly where? *}

lemma "\<not> (A \<and> B) \<longrightarrow> \<not> A \<or> \<not> B"

proof

  assume n: "\<not> (A \<and> B)"

  show "\<not> A \<or> \<not> B"

  proof (rule ccontr)

    assume nn: "\<not> (\<not> A \<or> \<not> B)"

    have "\<not> A"

    proof

      assume "A"

      have "\<not> B"

      proof

        assume "B"

        have "A \<and> B" ..

        with n show False ..

      qed

      hence "\<not> A \<or> \<not> B" ..

      with nn show False ..

    qed

    hence "\<not> A \<or> \<not> B" ..

    with nn show False ..

  qed

qed

text{*\noindent

Rule @{thm[source]ccontr} (``classical contradiction'') is

@{thm ccontr[no_vars]}.

Apart from demonstrating the strangeness of classical

arguments by contradiction, this example also introduces two new

abbreviations:

\begin{center}

\begin{tabular}{l@ {\quad=\quad}l}

\isakeyword{hence} & \isakeyword{then} \isakeyword{have} \\

\isakeyword{with}~\emph{facts} &

\isakeyword{from}~\emph{facts} @{text this}

\end{tabular}

\end{center}

*}

subsection{*Avoiding duplication*}

text{* So far our examples have been a bit unnatural: normally we want to

prove rules expressed with @{text"\<Longrightarrow>"}, not @{text"\<longrightarrow>"}. Here is an example:

*}

lemma "A \<and> B \<Longrightarrow> B \<and> A"

proof

  assume "A \<and> B" thus "B" ..

next

  assume "A \<and> B" thus "A" ..

qed

text{*\noindent The \isakeyword{proof} always works on the conclusion,

@{prop"B \<and> A"} in our case, thus selecting $\land$-introduction. Hence

we must show @{prop B} and @{prop A}; both are proved by

$\land$-elimination and the proofs are separated by \isakeyword{next}:

\begin{description}

\item[\isakeyword{next}] deals with multiple subgoals. For example,

when showing @{term"A \<and> B"} we need to show both @{term A} and @{term

B}.  Each subgoal is proved separately, in \emph{any} order. The

individual proofs are separated by \isakeyword{next}.  \footnote{Each

\isakeyword{show} must prove one of the pending subgoals.  If a

\isakeyword{show} matches multiple subgoals, e.g.\ if the subgoals

contain ?-variables, the first one is proved. Thus the order in which

the subgoals are proved can matter --- see

\S\ref{sec:CaseDistinction} for an example.}

Strictly speaking \isakeyword{next} is only required if the subgoals

are proved in different assumption contexts which need to be

separated, which is not the case above. For clarity we

have employed \isakeyword{next} anyway and will continue to do so.

\end{description}

This is all very well as long as formulae are small. Let us now look at some

devices to avoid repeating (possibly large) formulae. A very general method

is pattern matching: *}

lemma "large_A \<and> large_B \<Longrightarrow> large_B \<and> large_A"

      (is "?AB \<Longrightarrow> ?B \<and> ?A")

proof

  assume "?AB" thus "?B" ..

next

  assume "?AB" thus "?A" ..

qed

text{*\noindent Any formula may be followed by

@{text"("}\isakeyword{is}~\emph{pattern}@{text")"} which causes the pattern

to be matched against the formula, instantiating the @{text"?"}-variables in

the pattern. Subsequent uses of these variables in other terms causes

them to be replaced by the terms they stand for.

We can simplify things even more by stating the theorem by means of the

\isakeyword{assumes} and \isakeyword{shows} elements which allow direct

naming of assumptions: *}

lemma assumes AB: "large_A \<and> large_B"

  shows "large_B \<and> large_A" (is "?B \<and> ?A")

proof

  from AB show "?B" ..

next

  from AB show "?A" ..

qed

text{*\noindent Note the difference between @{text ?AB}, a term, and

@{text AB}, a fact.

Finally we want to start the proof with $\land$-elimination so we

don't have to perform it twice, as above. Here is a slick way to

achieve this: *}

lemma assumes AB: "large_A \<and> large_B"

  shows "large_B \<and> large_A" (is "?B \<and> ?A")

using AB

proof

  assume "?A" "?B" show ?thesis ..

qed

text{*\noindent Command \isakeyword{using} can appear before a proof

and adds further facts to those piped into the proof. Here @{text AB}

is the only such fact and it triggers $\land$-elimination. Another

frequent idiom is as follows:

\begin{center}

\isakeyword{from} \emph{major-facts}~

\isakeyword{show} \emph{proposition}~

\isakeyword{using} \emph{minor-facts}~

\emph{proof}

\end{center}

Sometimes it is necessary to suppress the implicit application of rules in a

\isakeyword{proof}. For example \isakeyword{show}~@{prop"A \<or> B"} would

trigger $\lor$-introduction, requiring us to prove @{prop A}. A simple

``@{text"-"}'' prevents this \emph{faux pas}: *}

lemma assumes AB: "A \<or> B" shows "B \<or> A"

proof -

  from AB show ?thesis

  proof

    assume A show ?thesis ..

  next

    assume B show ?thesis ..

  qed

qed

text{* Too many names can easily clutter a proof.  We already learned

about @{text this} as a means of avoiding explicit names. Another

handy device is to refer to a fact not by name but by contents: for

example, writing @{text "`A \<or> B`"} (enclosing the formula in back quotes)

refers to the fact @{text"A \<or> B"}

without the need to name it. Here is a simple example, a revised version

of the previous proof *}

lemma assumes "A \<or> B" shows "B \<or> A"

proof -

  from `A \<or> B` show ?thesis

(*<*)oops(*>*)

text{*\noindent which continues as before.

Clearly, this device of quoting facts by contents is only advisable

for small formulae. In such cases it is superior to naming because the

reader immediately sees what the fact is without needing to search for

it in the preceding proof text. *}

subsection{*Predicate calculus*}

text{* Command \isakeyword{fix} introduces new local variables into a

proof. The pair \isakeyword{fix}-\isakeyword{show} corresponds to @{text"\<And>"}

(the universal quantifier at the

meta-level) just like \isakeyword{assume}-\isakeyword{show} corresponds to

@{text"\<Longrightarrow>"}. Here is a sample proof, annotated with the rules that are

applied implicitly: *}

lemma assumes P: "\<forall>x. P x" shows "\<forall>x. P(f x)"

proof                       --"@{thm[source]allI}: @{thm allI}"

  fix a

  from P show "P(f a)" ..  --"@{thm[source]allE}: @{thm allE}"

qed

text{*\noindent Note that in the proof we have chosen to call the bound

variable @{term a} instead of @{term x} merely to show that the choice of

local names is irrelevant.

Next we look at @{text"\<exists>"} which is a bit more tricky.

*}

lemma assumes Pf: "\<exists>x. P(f x)" shows "\<exists>y. P y"

proof -

  from Pf show ?thesis

  proof              -- "@{thm[source]exE}: @{thm exE}"

    fix x

    assume "P(f x)"

    show ?thesis ..  -- "@{thm[source]exI}: @{thm exI}"

  qed

qed

text{*\noindent Explicit $\exists$-elimination as seen above can become

cumbersome in practice.  The derived Isar language element

\isakeyword{obtain} provides a more appealing form of generalised

existence reasoning: *}

lemma assumes Pf: "\<exists>x. P(f x)" shows "\<exists>y. P y"

proof -

  from Pf obtain x where "P(f x)" ..

  thus "\<exists>y. P y" ..

qed

text{*\noindent Note how the proof text follows the usual mathematical style

of concluding $P(x)$ from $\exists x. P(x)$, while carefully introducing $x$

as a new local variable.  Technically, \isakeyword{obtain} is similar to

\isakeyword{fix} and \isakeyword{assume} together with a soundness proof of

the elimination involved.

Here is a proof of a well known tautology.

Which rule is used where?  *}

lemma assumes ex: "\<exists>x. \<forall>y. P x y" shows "\<forall>y. \<exists>x. P x y"

proof

  fix y

  from ex obtain x where "\<forall>y. P x y" ..

  hence "P x y" ..

  thus "\<exists>x. P x y" ..

qed

subsection{*Making bigger steps*}

text{* So far we have confined ourselves to single step proofs. Of course

powerful automatic methods can be used just as well. Here is an example,

Cantor's theorem that there is no surjective function from a set to its

powerset: *}

theorem "\<exists>S. S \<notin> range (f :: 'a \<Rightarrow> 'a set)"

proof

  let ?S = "{x. x \<notin> f x}"

  show "?S \<notin> range f"

  proof

    assume "?S \<in> range f"

    then obtain y where "?S = f y" ..

    show False

    proof cases

      assume "y \<in> ?S"

      with `?S = f y` show False by blast

    next

      assume "y \<notin> ?S"

      with `?S = f y` show False by blast

    qed

  qed

qed

text{*\noindent

For a start, the example demonstrates two new constructs:

\begin{itemize}

\item \isakeyword{let} introduces an abbreviation for a term, in our case

the witness for the claim.

\item Proof by @{text"cases"} starts a proof by cases. Note that it remains

implicit what the two cases are: it is merely expected that the two subproofs

prove @{text"P \<Longrightarrow> ?thesis"} and @{text"\<not>P \<Longrightarrow> ?thesis"} (in that order)

for some @{term P}.

\end{itemize}

If you wonder how to \isakeyword{obtain} @{term y}:

via the predefined elimination rule @{thm rangeE[no_vars]}.

Method @{text blast} is used because the contradiction does not follow easily

by just a single rule. If you find the proof too cryptic for human

consumption, here is a more detailed version; the beginning up to

\isakeyword{obtain} stays unchanged. *}

theorem "\<exists>S. S \<notin> range (f :: 'a \<Rightarrow> 'a set)"

proof

  let ?S = "{x. x \<notin> f x}"

  show "?S \<notin> range f"

  proof

    assume "?S \<in> range f"

    then obtain y where "?S = f y" ..

    show False

    proof cases

      assume "y \<in> ?S"

      hence "y \<notin> f y"   by simp

      hence "y \<notin> ?S"    by(simp add: `?S = f y`)

      thus False         by contradiction

    next

      assume "y \<notin> ?S"

      hence "y \<in> f y"   by simp

      hence "y \<in> ?S"    by(simp add: `?S = f y`)

      thus False         by contradiction

    qed

  qed

qed

text{*\noindent Method @{text contradiction} succeeds if both $P$ and

$\neg P$ are among the assumptions and the facts fed into that step, in any order.

As it happens, Cantor's theorem can be proved automatically by best-first

search. Depth-first search would diverge, but best-first search successfully

navigates through the large search space:

*}

theorem "\<exists>S. S \<notin> range (f :: 'a \<Rightarrow> 'a set)"

by best

(* Of course this only works in the context of HOL's carefully

constructed introduction and elimination rules for set theory.*)

subsection{*Raw proof blocks*}

text{* Although we have shown how to employ powerful automatic methods like

@{text blast} to achieve bigger proof steps, there may still be the

tendency to use the default introduction and elimination rules to

decompose goals and facts. This can lead to very tedious proofs:

*}

(*<*)ML"set quick_and_dirty"(*>*)

lemma "\<forall>x y. A x y \<and> B x y \<longrightarrow> C x y"

proof

  fix x show "\<forall>y. A x y \<and> B x y \<longrightarrow> C x y"

  proof

    fix y show "A x y \<and> B x y \<longrightarrow> C x y"

    proof

      assume "A x y \<and> B x y"

      show "C x y" sorry

    qed

  qed

qed

text{*\noindent Since we are only interested in the decomposition and not the

actual proof, the latter has been replaced by

\isakeyword{sorry}. Command \isakeyword{sorry} proves anything but is

only allowed in quick and dirty mode, the default interactive mode. It

is very convenient for top down proof development.

Luckily we can avoid this step by step decomposition very easily: *}

lemma "\<forall>x y. A x y \<and> B x y \<longrightarrow> C x y"

proof -

  have "\<And>x y. \<lbrakk> A x y; B x y \<rbrakk> \<Longrightarrow> C x y"

  proof -

    fix x y assume "A x y" "B x y"

    show "C x y" sorry

  qed

  thus ?thesis by blast

qed

text{*\noindent

This can be simplified further by \emph{raw proof blocks}, i.e.\

proofs enclosed in braces: *}

lemma "\<forall>x y. A x y \<and> B x y \<longrightarrow> C x y"

proof -

  { fix x y assume "A x y" "B x y"

    have "C x y" sorry }

  thus ?thesis by blast

qed

text{*\noindent The result of the raw proof block is the same theorem

as above, namely @{prop"\<And>x y. \<lbrakk> A x y; B x y \<rbrakk> \<Longrightarrow> C x y"}.  Raw

proof blocks are like ordinary proofs except that they do not prove

some explicitly stated property but that the property emerges directly

out of the \isakeyword{fixe}s, \isakeyword{assume}s and

\isakeyword{have} in the block. Thus they again serve to avoid

duplication. Note that the conclusion of a raw proof block is stated with

\isakeyword{have} rather than \isakeyword{show} because it is not the

conclusion of some pending goal but some independent claim.

The general idea demonstrated in this subsection is very

important in Isar and distinguishes it from tactic-style proofs:

\begin{quote}\em

Do not manipulate the proof state into a particular form by applying

tactics but state the desired form explicitly and let the tactic verify

that from this form the original goal follows.

\end{quote}

This yields more readable and also more robust proofs.

\subsubsection{General case distinctions}

As an important application of raw proof blocks we show how to deal

with general case distinctions --- more specific kinds are treated in

\S\ref{sec:CaseDistinction}. Imagine that you would like to prove some

goal by distinguishing $n$ cases $P_1$, \dots, $P_n$. You show that

the $n$ cases are exhaustive (i.e.\ $P_1 \lor \dots \lor P_n$) and

that each case $P_i$ implies the goal. Taken together, this proves the

goal. The corresponding Isar proof pattern (for $n = 3$) is very handy:

*}

text_raw{*\renewcommand{\isamarkupcmt}[1]{#1}*}

(*<*)lemma "XXX"(*>*)

proof -

  have "P\<^isub>1 \<or> P\<^isub>2 \<or> P\<^isub>3" (*<*)sorry(*>*) -- {*\dots*}

  moreover

  { assume P\<^isub>1

    -- {*\dots*}

    have ?thesis (*<*)sorry(*>*) -- {*\dots*} }

  moreover

  { assume P\<^isub>2

    -- {*\dots*}

    have ?thesis (*<*)sorry(*>*) -- {*\dots*} }

  moreover

  { assume P\<^isub>3

    -- {*\dots*}

    have ?thesis (*<*)sorry(*>*) -- {*\dots*} }

  ultimately show ?thesis by blast

qed

text_raw{*\renewcommand{\isamarkupcmt}[1]{{\isastylecmt--- #1}}*}

subsection{*Further refinements*}

text{* This subsection discusses some further tricks that can make

life easier although they are not essential. *}

subsubsection{*\isakeyword{and}*}

text{* Propositions (following \isakeyword{assume} etc) may but need not be

separated by \isakeyword{and}. This is not just for readability

(\isakeyword{from} \isa{A} \isakeyword{and} \isa{B} looks nicer than

\isakeyword{from} \isa{A} \isa{B}) but for structuring lists of propositions

into possibly named blocks. In

\begin{center}

\isakeyword{assume} \isa{A:} $A_1$ $A_2$ \isakeyword{and} \isa{B:} $A_3$

\isakeyword{and} $A_4$

\end{center}

label \isa{A} refers to the list of propositions $A_1$ $A_2$ and

label \isa{B} to $A_3$. *}

subsubsection{*\isakeyword{note}*}

text{* If you want to remember intermediate fact(s) that cannot be

named directly, use \isakeyword{note}. For example the result of raw

proof block can be named by following it with

\isakeyword{note}~@{text"some_name = this"}.  As a side effect,

@{text this} is set to the list of facts on the right-hand side. You

can also say @{text"note some_fact"}, which simply sets @{text this},

i.e.\ recalls @{text"some_fact"}, e.g.\ in a \isakeyword{moreover} sequence. *}

subsubsection{*\isakeyword{fixes}*}

text{* Sometimes it is necessary to decorate a proposition with type

constraints, as in Cantor's theorem above. These type constraints tend

to make the theorem less readable. The situation can be improved a

little by combining the type constraint with an outer @{text"\<And>"}: *}

theorem "\<And>f :: 'a \<Rightarrow> 'a set. \<exists>S. S \<notin> range f"

(*<*)oops(*>*)

text{*\noindent However, now @{term f} is bound and we need a

\isakeyword{fix}~\isa{f} in the proof before we can refer to @{term f}.

This is avoided by \isakeyword{fixes}: *}

theorem fixes f :: "'a \<Rightarrow> 'a set" shows "\<exists>S. S \<notin> range f"

(*<*)oops(*>*)

text{* \noindent

Even better, \isakeyword{fixes} allows to introduce concrete syntax locally:*}

lemma comm_mono:

  fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix ">" 60) and

       f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   (infixl "++" 70)

  assumes comm: "\<And>x y::'a. x ++ y = y ++ x" and

          mono: "\<And>x y z::'a. x > y \<Longrightarrow> x ++ z > y ++ z"

  shows "x > y \<Longrightarrow> z ++ x > z ++ y"

by(simp add: comm mono)

text{*\noindent The concrete syntax is dropped at the end of the proof and the

theorem becomes @{thm[display,margin=60]comm_mono}

\tweakskip *}

subsubsection{*\isakeyword{obtain}*}

text{* The \isakeyword{obtain} construct can introduce multiple

witnesses and propositions as in the following proof fragment:*}

lemma assumes A: "\<exists>x y. P x y \<and> Q x y" shows "R"

proof -

  from A obtain x y where P: "P x y" and Q: "Q x y"  by blast

(*<*)oops(*>*)

text{* Remember also that one does not even need to start with a formula

containing @{text"\<exists>"} as we saw in the proof of Cantor's theorem.

*}

subsubsection{*Combining proof styles*}

text{* Finally, whole ``scripts'' (tactic-based proofs in the style of

\cite{LNCS2283}) may appear in the leaves of the proof tree, although this is

best avoided.  Here is a contrived example: *}

lemma "A \<longrightarrow> (A \<longrightarrow> B) \<longrightarrow> B"

proof

  assume a: "A"

  show "(A \<longrightarrow>B) \<longrightarrow> B"

    apply(rule impI)

    apply(erule impE)

    apply(rule a)

    apply assumption

    done

qed

text{*\noindent You may need to resort to this technique if an

automatic step fails to prove the desired proposition.

When converting a proof from tactic-style into Isar you can proceed

in a top-down manner: parts of the proof can be left in script form

while the outer structure is already expressed in Isar. *}

(*<*)end(*>*)