src/HOL/Isar_examples/BasicLogic.thy
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improved theory reference in comment
     1 (*  Title:      HOL/Isar_examples/BasicLogic.thy

     2     ID:         $Id$

     3     Author:     Markus Wenzel, TU Muenchen

     4

     5 Basic propositional and quantifier reasoning.

     6 *)

     7

     8 header {* Basic logical reasoning *}

     9

    10 theory BasicLogic = Main:

    11

    12

    13 subsection {* Pure backward reasoning *}

    14

    15 text {*

    16  In order to get a first idea of how Isabelle/Isar proof documents may

    17  look like, we consider the propositions $I$, $K$, and $S$.  The

    18  following (rather explicit) proofs should require little extra

    19  explanations.

    20 *}

    21

    22 lemma I: "A --> A"

    23 proof

    24   assume A

    25   show A by assumption

    26 qed

    27

    28 lemma K: "A --> B --> A"

    29 proof

    30   assume A

    31   show "B --> A"

    32   proof

    33     show A by assumption

    34   qed

    35 qed

    36

    37 lemma S: "(A --> B --> C) --> (A --> B) --> A --> C"

    38 proof

    39   assume "A --> B --> C"

    40   show "(A --> B) --> A --> C"

    41   proof

    42     assume "A --> B"

    43     show "A --> C"

    44     proof

    45       assume A

    46       show C

    47       proof (rule mp)

    48 	show "B --> C" by (rule mp)

    49         show B by (rule mp)

    50       qed

    51     qed

    52   qed

    53 qed

    54

    55 text {*

    56  Isar provides several ways to fine-tune the reasoning, avoiding

    57  excessive detail.  Several abbreviated language elements are

    58  available, enabling the writer to express proofs in a more concise

    59  way, even without referring to any automated proof tools yet.

    60

    61  First of all, proof by assumption may be abbreviated as a single dot.

    62 *}

    63

    64 lemma "A --> A"

    65 proof

    66   assume A

    67   show A .

    68 qed

    69

    70 text {*

    71  In fact, concluding any (sub-)proof already involves solving any

    72  remaining goals by assumption\footnote{This is not a completely

    73  trivial operation, as proof by assumption may involve full

    74  higher-order unification.}.  Thus we may skip the rather vacuous body

    75  of the above proof as well.

    76 *}

    77

    78 lemma "A --> A"

    79 proof

    80 qed

    81

    82 text {*

    83  Note that the \isacommand{proof} command refers to the $\idt{rule}$

    84  method (without arguments) by default.  Thus it implicitly applies a

    85  single rule, as determined from the syntactic form of the statements

    86  involved.  The \isacommand{by} command abbreviates any proof with

    87  empty body, so the proof may be further pruned.

    88 *}

    89

    90 lemma "A --> A"

    91   by rule

    92

    93 text {*

    94  Proof by a single rule may be abbreviated as double-dot.

    95 *}

    96

    97 lemma "A --> A" ..

    98

    99 text {*

   100  Thus we have arrived at an adequate representation of the proof of a

   101  tautology that holds by a single standard rule.\footnote{Apparently,

   102  the rule here is implication introduction.}

   103 *}

   104

   105 text {*

   106  Let us also reconsider $K$.  Its statement is composed of iterated

   107  connectives.  Basic decomposition is by a single rule at a time,

   108  which is why our first version above was by nesting two proofs.

   109

   110  The $\idt{intro}$ proof method repeatedly decomposes a goal's

   111  conclusion.\footnote{The dual method is $\idt{elim}$, acting on a

   112  goal's premises.}

   113 *}

   114

   115 lemma "A --> B --> A"

   116 proof intro

   117   assume A

   118   show A .

   119 qed

   120

   121 text {*

   122  Again, the body may be collapsed.

   123 *}

   124

   125 lemma "A --> B --> A"

   126   by intro

   127

   128 text {*

   129  Just like $\idt{rule}$, the $\idt{intro}$ and $\idt{elim}$ proof

   130  methods pick standard structural rules, in case no explicit arguments

   131  are given.  While implicit rules are usually just fine for single

   132  rule application, this may go too far with iteration.  Thus in

   133  practice, $\idt{intro}$ and $\idt{elim}$ would be typically

   134  restricted to certain structures by giving a few rules only, e.g.\

   135  \isacommand{proof}~($\idt{intro}$~\name{impI}~\name{allI}) to strip

   136  implications and universal quantifiers.

   137

   138  Such well-tuned iterated decomposition of certain structures is the

   139  prime application of $\idt{intro}$ and $\idt{elim}$.  In contrast,

   140  terminal steps that solve a goal completely are usually performed by

   141  actual automated proof methods (such as

   142  \isacommand{by}~$\idt{blast}$).

   143 *}

   144

   145

   146 subsection {* Variations of backward vs.\ forward reasoning *}

   147

   148 text {*

   149  Certainly, any proof may be performed in backward-style only.  On the

   150  other hand, small steps of reasoning are often more naturally

   151  expressed in forward-style.  Isar supports both backward and forward

   152  reasoning as a first-class concept.  In order to demonstrate the

   153  difference, we consider several proofs of $A \conj B \impl B \conj   154 A$.

   155

   156  The first version is purely backward.

   157 *}

   158

   159 lemma "A & B --> B & A"

   160 proof

   161   assume "A & B"

   162   show "B & A"

   163   proof

   164     show B by (rule conjunct2)

   165     show A by (rule conjunct1)

   166   qed

   167 qed

   168

   169 text {*

   170  Above, the $\idt{conjunct}_{1/2}$ projection rules had to be named

   171  explicitly, since the goals $B$ and $A$ did not provide any

   172  structural clue.  This may be avoided using \isacommand{from} to

   173  focus on $\idt{prems}$ (i.e.\ the $A \conj B$ assumption) as the

   174  current facts, enabling the use of double-dot proofs.  Note that

   175  \isacommand{from} already does forward-chaining, involving the

   176  \name{conjE} rule here.

   177 *}

   178

   179 lemma "A & B --> B & A"

   180 proof

   181   assume "A & B"

   182   show "B & A"

   183   proof

   184     from prems show B ..

   185     from prems show A ..

   186   qed

   187 qed

   188

   189 text {*

   190  In the next version, we move the forward step one level upwards.

   191  Forward-chaining from the most recent facts is indicated by the

   192  \isacommand{then} command.  Thus the proof of $B \conj A$ from $A   193 \conj B$ actually becomes an elimination, rather than an

   194  introduction.  The resulting proof structure directly corresponds to

   195  that of the $\name{conjE}$ rule, including the repeated goal

   196  proposition that is abbreviated as $\var{thesis}$ below.

   197 *}

   198

   199 lemma "A & B --> B & A"

   200 proof

   201   assume "A & B"

   202   then show "B & A"

   203   proof                    -- {* rule \name{conjE} of $A \conj B$ *}

   204     assume A B

   205     show ?thesis ..       -- {* rule \name{conjI} of $B \conj A$ *}

   206   qed

   207 qed

   208

   209 text {*

   210  In the subsequent version we flatten the structure of the main body

   211  by doing forward reasoning all the time.  Only the outermost

   212  decomposition step is left as backward.

   213 *}

   214

   215 lemma "A & B --> B & A"

   216 proof

   217   assume ab: "A & B"

   218   from ab have a: A ..

   219   from ab have b: B ..

   220   from b a show "B & A" ..

   221 qed

   222

   223 text {*

   224  We can still push forward-reasoning a bit further, even at the risk

   225  of getting ridiculous.  Note that we force the initial proof step to

   226  do nothing here, by referring to the -'' proof method.

   227 *}

   228

   229 lemma "A & B --> B & A"

   230 proof -

   231   {

   232     assume ab: "A & B"

   233     from ab have a: A ..

   234     from ab have b: B ..

   235     from b a have "B & A" ..

   236   }

   237   thus ?thesis ..         -- {* rule \name{impI} *}

   238 qed

   239

   240 text {*

   241  \medskip With these examples we have shifted through a whole range

   242  from purely backward to purely forward reasoning.  Apparently, in the

   243  extreme ends we get slightly ill-structured proofs, which also

   244  require much explicit naming of either rules (backward) or local

   245  facts (forward).

   246

   247  The general lesson learned here is that good proof style would

   248  achieve just the \emph{right} balance of top-down backward

   249  decomposition, and bottom-up forward composition.  In general, there

   250  is no single best way to arrange some pieces of formal reasoning, of

   251  course.  Depending on the actual applications, the intended audience

   252  etc., rules (and methods) on the one hand vs.\ facts on the other

   253  hand have to be emphasized in an appropriate way.  This requires the

   254  proof writer to develop good taste, and some practice, of course.

   255 *}

   256

   257 text {*

   258  For our example the most appropriate way of reasoning is probably the

   259  middle one, with conjunction introduction done after elimination.

   260  This reads even more concisely using \isacommand{thus}, which

   261  abbreviates \isacommand{then}~\isacommand{show}.\footnote{In the same

   262  vein, \isacommand{hence} abbreviates

   263  \isacommand{then}~\isacommand{have}.}

   264 *}

   265

   266 lemma "A & B --> B & A"

   267 proof

   268   assume "A & B"

   269   thus "B & A"

   270   proof

   271     assume A B

   272     show ?thesis ..

   273   qed

   274 qed

   275

   276

   277

   278 subsection {* A few examples from Introduction to Isabelle'' *}

   279

   280 text {*

   281  We rephrase some of the basic reasoning examples of

   282  \cite{isabelle-intro}, using HOL rather than FOL.

   283 *}

   284

   285 subsubsection {* A propositional proof *}

   286

   287 text {*

   288  We consider the proposition $P \disj P \impl P$.  The proof below

   289  involves forward-chaining from $P \disj P$, followed by an explicit

   290  case-analysis on the two \emph{identical} cases.

   291 *}

   292

   293 lemma "P | P --> P"

   294 proof

   295   assume "P | P"

   296   thus P

   297   proof                    -- {*

   298     rule \name{disjE}: \smash{$\infer{C}{A \disj B & \infer*{C}{[A]} & \infer*{C}{[B]}}$}

   299   *}

   300     assume P show P .

   301   next

   302     assume P show P .

   303   qed

   304 qed

   305

   306 text {*

   307  Case splits are \emph{not} hardwired into the Isar language as a

   308  special feature.  The \isacommand{next} command used to separate the

   309  cases above is just a short form of managing block structure.

   310

   311  \medskip In general, applying proof methods may split up a goal into

   312  separate cases'', i.e.\ new subgoals with individual local

   313  assumptions.  The corresponding proof text typically mimics this by

   314  establishing results in appropriate contexts, separated by blocks.

   315

   316  In order to avoid too much explicit parentheses, the Isar system

   317  implicitly opens an additional block for any new goal, the

   318  \isacommand{next} statement then closes one block level, opening a

   319  new one.  The resulting behavior is what one would expect from

   320  separating cases, only that it is more flexible.  E.g.\ an induction

   321  base case (which does not introduce local assumptions) would

   322  \emph{not} require \isacommand{next} to separate the subsequent step

   323  case.

   324

   325  \medskip In our example the situation is even simpler, since the two

   326  cases actually coincide.  Consequently the proof may be rephrased as

   327  follows.

   328 *}

   329

   330 lemma "P | P --> P"

   331 proof

   332   assume "P | P"

   333   thus P

   334   proof

   335     assume P

   336     show P .

   337     show P .

   338   qed

   339 qed

   340

   341 text {*

   342  Again, the rather vacuous body of the proof may be collapsed.  Thus

   343  the case analysis degenerates into two assumption steps, which are

   344  implicitly performed when concluding the single rule step of the

   345  double-dot proof as follows.

   346 *}

   347

   348 lemma "P | P --> P"

   349 proof

   350   assume "P | P"

   351   thus P ..

   352 qed

   353

   354

   355 subsubsection {* A quantifier proof *}

   356

   357 text {*

   358  To illustrate quantifier reasoning, let us prove $(\ex x P \ap (f \ap   359 x)) \impl (\ex x P \ap x)$.  Informally, this holds because any $a$

   360  with $P \ap (f \ap a)$ may be taken as a witness for the second

   361  existential statement.

   362

   363  The first proof is rather verbose, exhibiting quite a lot of

   364  (redundant) detail.  It gives explicit rules, even with some

   365  instantiation.  Furthermore, we encounter two new language elements:

   366  the \isacommand{fix} command augments the context by some new

   367  arbitrary, but fixed'' element; the \isacommand{is} annotation

   368  binds term abbreviations by higher-order pattern matching.

   369 *}

   370

   371 lemma "(EX x. P (f x)) --> (EX y. P y)"

   372 proof

   373   assume "EX x. P (f x)"

   374   thus "EX y. P y"

   375   proof (rule exE)             -- {*

   376     rule \name{exE}: \smash{$\infer{B}{\ex x A(x) & \infer*{B}{[A(x)]_x}}$}

   377   *}

   378     fix a

   379     assume "P (f a)" (is "P ?witness")

   380     show ?thesis by (rule exI [of P ?witness])

   381   qed

   382 qed

   383

   384 text {*

   385  While explicit rule instantiation may occasionally improve

   386  readability of certain aspects of reasoning, it is usually quite

   387  redundant.  Above, the basic proof outline gives already enough

   388  structural clues for the system to infer both the rules and their

   389  instances (by higher-order unification).  Thus we may as well prune

   390  the text as follows.

   391 *}

   392

   393 lemma "(EX x. P (f x)) --> (EX y. P y)"

   394 proof

   395   assume "EX x. P (f x)"

   396   thus "EX y. P y"

   397   proof

   398     fix a

   399     assume "P (f a)"

   400     show ?thesis ..

   401   qed

   402 qed

   403

   404 text {*

   405  Explicit $\exists$-elimination as seen above can become quite

   406  cumbersome in practice.  The derived Isar language element

   407  \isakeyword{obtain}'' provides a more handsome way to do

   408  generalized existence reasoning.

   409 *}

   410

   411 lemma "(EX x. P (f x)) --> (EX y. P y)"

   412 proof

   413   assume "EX x. P (f x)"

   414   then obtain a where "P (f a)" ..

   415   thus "EX y. P y" ..

   416 qed

   417

   418 text {*

   419  Technically, \isakeyword{obtain} is similar to \isakeyword{fix} and

   420  \isakeyword{assume} together with a soundness proof of the

   421  elimination involved.  Thus it behaves similar to any other forward

   422  proof element.  Also note that due to the nature of general existence

   423  reasoning involved here, any result exported from the context of an

   424  \isakeyword{obtain} statement may \emph{not} refer to the parameters

   425  introduced there.

   426 *}

   427

   428

   429

   430 subsubsection {* Deriving rules in Isabelle *}

   431

   432 text {*

   433  We derive the conjunction elimination rule from the corresponding

   434  projections.  The proof is quite straight-forward, since

   435  Isabelle/Isar supports non-atomic goals and assumptions fully

   436  transparently.

   437 *}

   438

   439 theorem conjE: "A & B ==> (A ==> B ==> C) ==> C"

   440 proof -

   441   assume "A & B"

   442   assume r: "A ==> B ==> C"

   443   show C

   444   proof (rule r)

   445     show A by (rule conjunct1)

   446     show B by (rule conjunct2)

   447   qed

   448 qed

   449

   450 text {*

   451  Note that classic Isabelle handles higher rules in a slightly

   452  different way.  The tactic script as given in \cite{isabelle-intro}

   453  for the same example of \name{conjE} depends on the primitive

   454  \texttt{goal} command to decompose the rule into premises and

   455  conclusion.  The actual result would then emerge by discharging of

   456  the context at \texttt{qed} time.

   457 *}

   458

   459 end