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src/HOL/Isar_Examples/Basic_Logic.thy

changeset 37671 | fa53d267dab3 |

parent 33026 | 8f35633c4922 |

child 55640 | abc140f21caa |

1.1 --- a/src/HOL/Isar_Examples/Basic_Logic.thy Thu Jul 01 14:32:57 2010 +0200 1.2 +++ b/src/HOL/Isar_Examples/Basic_Logic.thy Thu Jul 01 18:31:46 2010 +0200 1.3 @@ -13,12 +13,10 @@ 1.4 1.5 subsection {* Pure backward reasoning *} 1.6 1.7 -text {* 1.8 - In order to get a first idea of how Isabelle/Isar proof documents 1.9 - may look like, we consider the propositions @{text I}, @{text K}, 1.10 - and @{text S}. The following (rather explicit) proofs should 1.11 - require little extra explanations. 1.12 -*} 1.13 +text {* In order to get a first idea of how Isabelle/Isar proof 1.14 + documents may look like, we consider the propositions @{text I}, 1.15 + @{text K}, and @{text S}. The following (rather explicit) proofs 1.16 + should require little extra explanations. *} 1.17 1.18 lemma I: "A --> A" 1.19 proof 1.20 @@ -53,15 +51,14 @@ 1.21 qed 1.22 qed 1.23 1.24 -text {* 1.25 - Isar provides several ways to fine-tune the reasoning, avoiding 1.26 - excessive detail. Several abbreviated language elements are 1.27 - available, enabling the writer to express proofs in a more concise 1.28 - way, even without referring to any automated proof tools yet. 1.29 +text {* Isar provides several ways to fine-tune the reasoning, 1.30 + avoiding excessive detail. Several abbreviated language elements 1.31 + are available, enabling the writer to express proofs in a more 1.32 + concise way, even without referring to any automated proof tools 1.33 + yet. 1.34 1.35 First of all, proof by assumption may be abbreviated as a single 1.36 - dot. 1.37 -*} 1.38 + dot. *} 1.39 1.40 lemma "A --> A" 1.41 proof 1.42 @@ -69,51 +66,42 @@ 1.43 show A by fact+ 1.44 qed 1.45 1.46 -text {* 1.47 - In fact, concluding any (sub-)proof already involves solving any 1.48 - remaining goals by assumption\footnote{This is not a completely 1.49 +text {* In fact, concluding any (sub-)proof already involves solving 1.50 + any remaining goals by assumption\footnote{This is not a completely 1.51 trivial operation, as proof by assumption may involve full 1.52 higher-order unification.}. Thus we may skip the rather vacuous 1.53 - body of the above proof as well. 1.54 -*} 1.55 + body of the above proof as well. *} 1.56 1.57 lemma "A --> A" 1.58 proof 1.59 qed 1.60 1.61 -text {* 1.62 - Note that the \isacommand{proof} command refers to the @{text rule} 1.63 - method (without arguments) by default. Thus it implicitly applies a 1.64 - single rule, as determined from the syntactic form of the statements 1.65 - involved. The \isacommand{by} command abbreviates any proof with 1.66 - empty body, so the proof may be further pruned. 1.67 -*} 1.68 +text {* Note that the \isacommand{proof} command refers to the @{text 1.69 + rule} method (without arguments) by default. Thus it implicitly 1.70 + applies a single rule, as determined from the syntactic form of the 1.71 + statements involved. The \isacommand{by} command abbreviates any 1.72 + proof with empty body, so the proof may be further pruned. *} 1.73 1.74 lemma "A --> A" 1.75 by rule 1.76 1.77 -text {* 1.78 - Proof by a single rule may be abbreviated as double-dot. 1.79 -*} 1.80 +text {* Proof by a single rule may be abbreviated as double-dot. *} 1.81 1.82 lemma "A --> A" .. 1.83 1.84 -text {* 1.85 - Thus we have arrived at an adequate representation of the proof of a 1.86 - tautology that holds by a single standard rule.\footnote{Apparently, 1.87 - the rule here is implication introduction.} 1.88 -*} 1.89 +text {* Thus we have arrived at an adequate representation of the 1.90 + proof of a tautology that holds by a single standard 1.91 + rule.\footnote{Apparently, the rule here is implication 1.92 + introduction.} *} 1.93 1.94 -text {* 1.95 - Let us also reconsider @{text K}. Its statement is composed of 1.96 - iterated connectives. Basic decomposition is by a single rule at a 1.97 - time, which is why our first version above was by nesting two 1.98 +text {* Let us also reconsider @{text K}. Its statement is composed 1.99 + of iterated connectives. Basic decomposition is by a single rule at 1.100 + a time, which is why our first version above was by nesting two 1.101 proofs. 1.102 1.103 The @{text intro} proof method repeatedly decomposes a goal's 1.104 conclusion.\footnote{The dual method is @{text elim}, acting on a 1.105 - goal's premises.} 1.106 -*} 1.107 + goal's premises.} *} 1.108 1.109 lemma "A --> B --> A" 1.110 proof (intro impI) 1.111 @@ -121,16 +109,13 @@ 1.112 show A by fact 1.113 qed 1.114 1.115 -text {* 1.116 - Again, the body may be collapsed. 1.117 -*} 1.118 +text {* Again, the body may be collapsed. *} 1.119 1.120 lemma "A --> B --> A" 1.121 by (intro impI) 1.122 1.123 -text {* 1.124 - Just like @{text rule}, the @{text intro} and @{text elim} proof 1.125 - methods pick standard structural rules, in case no explicit 1.126 +text {* Just like @{text rule}, the @{text intro} and @{text elim} 1.127 + proof methods pick standard structural rules, in case no explicit 1.128 arguments are given. While implicit rules are usually just fine for 1.129 single rule application, this may go too far with iteration. Thus 1.130 in practice, @{text intro} and @{text elim} would be typically 1.131 @@ -142,21 +127,18 @@ 1.132 prime application of @{text intro} and @{text elim}. In contrast, 1.133 terminal steps that solve a goal completely are usually performed by 1.134 actual automated proof methods (such as \isacommand{by}~@{text 1.135 - blast}. 1.136 -*} 1.137 + blast}. *} 1.138 1.139 1.140 subsection {* Variations of backward vs.\ forward reasoning *} 1.141 1.142 -text {* 1.143 - Certainly, any proof may be performed in backward-style only. On 1.144 - the other hand, small steps of reasoning are often more naturally 1.145 +text {* Certainly, any proof may be performed in backward-style only. 1.146 + On the other hand, small steps of reasoning are often more naturally 1.147 expressed in forward-style. Isar supports both backward and forward 1.148 reasoning as a first-class concept. In order to demonstrate the 1.149 difference, we consider several proofs of @{text "A \<and> B \<longrightarrow> B \<and> A"}. 1.150 1.151 - The first version is purely backward. 1.152 -*} 1.153 + The first version is purely backward. *} 1.154 1.155 lemma "A & B --> B & A" 1.156 proof 1.157 @@ -168,14 +150,13 @@ 1.158 qed 1.159 qed 1.160 1.161 -text {* 1.162 - Above, the @{text "conjunct_1/2"} projection rules had to be named 1.163 - explicitly, since the goals @{text B} and @{text A} did not provide 1.164 - any structural clue. This may be avoided using \isacommand{from} to 1.165 - focus on the @{text "A \<and> B"} assumption as the current facts, 1.166 - enabling the use of double-dot proofs. Note that \isacommand{from} 1.167 - already does forward-chaining, involving the \name{conjE} rule here. 1.168 -*} 1.169 +text {* Above, the @{text "conjunct_1/2"} projection rules had to be 1.170 + named explicitly, since the goals @{text B} and @{text A} did not 1.171 + provide any structural clue. This may be avoided using 1.172 + \isacommand{from} to focus on the @{text "A \<and> B"} assumption as the 1.173 + current facts, enabling the use of double-dot proofs. Note that 1.174 + \isacommand{from} already does forward-chaining, involving the 1.175 + @{text conjE} rule here. *} 1.176 1.177 lemma "A & B --> B & A" 1.178 proof 1.179 @@ -187,15 +168,13 @@ 1.180 qed 1.181 qed 1.182 1.183 -text {* 1.184 - In the next version, we move the forward step one level upwards. 1.185 - Forward-chaining from the most recent facts is indicated by the 1.186 - \isacommand{then} command. Thus the proof of @{text "B \<and> A"} from 1.187 - @{text "A \<and> B"} actually becomes an elimination, rather than an 1.188 +text {* In the next version, we move the forward step one level 1.189 + upwards. Forward-chaining from the most recent facts is indicated 1.190 + by the \isacommand{then} command. Thus the proof of @{text "B \<and> A"} 1.191 + from @{text "A \<and> B"} actually becomes an elimination, rather than an 1.192 introduction. The resulting proof structure directly corresponds to 1.193 that of the @{text conjE} rule, including the repeated goal 1.194 - proposition that is abbreviated as @{text ?thesis} below. 1.195 -*} 1.196 + proposition that is abbreviated as @{text ?thesis} below. *} 1.197 1.198 lemma "A & B --> B & A" 1.199 proof 1.200 @@ -207,11 +186,9 @@ 1.201 qed 1.202 qed 1.203 1.204 -text {* 1.205 - In the subsequent version we flatten the structure of the main body 1.206 - by doing forward reasoning all the time. Only the outermost 1.207 - decomposition step is left as backward. 1.208 -*} 1.209 +text {* In the subsequent version we flatten the structure of the main 1.210 + body by doing forward reasoning all the time. Only the outermost 1.211 + decomposition step is left as backward. *} 1.212 1.213 lemma "A & B --> B & A" 1.214 proof 1.215 @@ -221,11 +198,9 @@ 1.216 from `B` `A` show "B & A" .. 1.217 qed 1.218 1.219 -text {* 1.220 - We can still push forward-reasoning a bit further, even at the risk 1.221 - of getting ridiculous. Note that we force the initial proof step to 1.222 - do nothing here, by referring to the ``-'' proof method. 1.223 -*} 1.224 +text {* We can still push forward-reasoning a bit further, even at the 1.225 + risk of getting ridiculous. Note that we force the initial proof 1.226 + step to do nothing here, by referring to the ``-'' proof method. *} 1.227 1.228 lemma "A & B --> B & A" 1.229 proof - 1.230 @@ -235,15 +210,14 @@ 1.231 from `A & B` have B .. 1.232 from `B` `A` have "B & A" .. 1.233 } 1.234 - then show ?thesis .. -- {* rule \name{impI} *} 1.235 + then show ?thesis .. -- {* rule @{text impI} *} 1.236 qed 1.237 1.238 -text {* 1.239 - \medskip With these examples we have shifted through a whole range 1.240 - from purely backward to purely forward reasoning. Apparently, in 1.241 - the extreme ends we get slightly ill-structured proofs, which also 1.242 - require much explicit naming of either rules (backward) or local 1.243 - facts (forward). 1.244 +text {* \medskip With these examples we have shifted through a whole 1.245 + range from purely backward to purely forward reasoning. Apparently, 1.246 + in the extreme ends we get slightly ill-structured proofs, which 1.247 + also require much explicit naming of either rules (backward) or 1.248 + local facts (forward). 1.249 1.250 The general lesson learned here is that good proof style would 1.251 achieve just the \emph{right} balance of top-down backward 1.252 @@ -252,14 +226,11 @@ 1.253 course. Depending on the actual applications, the intended audience 1.254 etc., rules (and methods) on the one hand vs.\ facts on the other 1.255 hand have to be emphasized in an appropriate way. This requires the 1.256 - proof writer to develop good taste, and some practice, of course. 1.257 -*} 1.258 + proof writer to develop good taste, and some practice, of course. *} 1.259 1.260 -text {* 1.261 - For our example the most appropriate way of reasoning is probably 1.262 - the middle one, with conjunction introduction done after 1.263 - elimination. 1.264 -*} 1.265 +text {* For our example the most appropriate way of reasoning is 1.266 + probably the middle one, with conjunction introduction done after 1.267 + elimination. *} 1.268 1.269 lemma "A & B --> B & A" 1.270 proof 1.271 @@ -275,18 +246,15 @@ 1.272 1.273 subsection {* A few examples from ``Introduction to Isabelle'' *} 1.274 1.275 -text {* 1.276 - We rephrase some of the basic reasoning examples of 1.277 - \cite{isabelle-intro}, using HOL rather than FOL. 1.278 -*} 1.279 +text {* We rephrase some of the basic reasoning examples of 1.280 + \cite{isabelle-intro}, using HOL rather than FOL. *} 1.281 + 1.282 1.283 subsubsection {* A propositional proof *} 1.284 1.285 -text {* 1.286 - We consider the proposition @{text "P \<or> P \<longrightarrow> P"}. The proof below 1.287 - involves forward-chaining from @{text "P \<or> P"}, followed by an 1.288 - explicit case-analysis on the two \emph{identical} cases. 1.289 -*} 1.290 +text {* We consider the proposition @{text "P \<or> P \<longrightarrow> P"}. The proof 1.291 + below involves forward-chaining from @{text "P \<or> P"}, followed by an 1.292 + explicit case-analysis on the two \emph{identical} cases. *} 1.293 1.294 lemma "P | P --> P" 1.295 proof 1.296 @@ -301,10 +269,9 @@ 1.297 qed 1.298 qed 1.299 1.300 -text {* 1.301 - Case splits are \emph{not} hardwired into the Isar language as a 1.302 - special feature. The \isacommand{next} command used to separate the 1.303 - cases above is just a short form of managing block structure. 1.304 +text {* Case splits are \emph{not} hardwired into the Isar language as 1.305 + a special feature. The \isacommand{next} command used to separate 1.306 + the cases above is just a short form of managing block structure. 1.307 1.308 \medskip In general, applying proof methods may split up a goal into 1.309 separate ``cases'', i.e.\ new subgoals with individual local 1.310 @@ -322,8 +289,7 @@ 1.311 1.312 \medskip In our example the situation is even simpler, since the two 1.313 cases actually coincide. Consequently the proof may be rephrased as 1.314 - follows. 1.315 -*} 1.316 + follows. *} 1.317 1.318 lemma "P | P --> P" 1.319 proof 1.320 @@ -336,12 +302,10 @@ 1.321 qed 1.322 qed 1.323 1.324 -text {* 1.325 - Again, the rather vacuous body of the proof may be collapsed. Thus 1.326 - the case analysis degenerates into two assumption steps, which are 1.327 - implicitly performed when concluding the single rule step of the 1.328 - double-dot proof as follows. 1.329 -*} 1.330 +text {* Again, the rather vacuous body of the proof may be collapsed. 1.331 + Thus the case analysis degenerates into two assumption steps, which 1.332 + are implicitly performed when concluding the single rule step of the 1.333 + double-dot proof as follows. *} 1.334 1.335 lemma "P | P --> P" 1.336 proof 1.337 @@ -352,26 +316,24 @@ 1.338 1.339 subsubsection {* A quantifier proof *} 1.340 1.341 -text {* 1.342 - To illustrate quantifier reasoning, let us prove @{text "(\<exists>x. P (f 1.343 - x)) \<longrightarrow> (\<exists>y. P y)"}. Informally, this holds because any @{text a} 1.344 - with @{text "P (f a)"} may be taken as a witness for the second 1.345 - existential statement. 1.346 +text {* To illustrate quantifier reasoning, let us prove @{text 1.347 + "(\<exists>x. P (f x)) \<longrightarrow> (\<exists>y. P y)"}. Informally, this holds because any 1.348 + @{text a} with @{text "P (f a)"} may be taken as a witness for the 1.349 + second existential statement. 1.350 1.351 The first proof is rather verbose, exhibiting quite a lot of 1.352 (redundant) detail. It gives explicit rules, even with some 1.353 instantiation. Furthermore, we encounter two new language elements: 1.354 the \isacommand{fix} command augments the context by some new 1.355 ``arbitrary, but fixed'' element; the \isacommand{is} annotation 1.356 - binds term abbreviations by higher-order pattern matching. 1.357 -*} 1.358 + binds term abbreviations by higher-order pattern matching. *} 1.359 1.360 lemma "(EX x. P (f x)) --> (EX y. P y)" 1.361 proof 1.362 assume "EX x. P (f x)" 1.363 then show "EX y. P y" 1.364 proof (rule exE) -- {* 1.365 - rule \name{exE}: \smash{$\infer{B}{\ex x A(x) & \infer*{B}{[A(x)]_x}}$} 1.366 + rule @{text exE}: \smash{$\infer{B}{\ex x A(x) & \infer*{B}{[A(x)]_x}}$} 1.367 *} 1.368 fix a 1.369 assume "P (f a)" (is "P ?witness") 1.370 @@ -379,14 +341,12 @@ 1.371 qed 1.372 qed 1.373 1.374 -text {* 1.375 - While explicit rule instantiation may occasionally improve 1.376 +text {* While explicit rule instantiation may occasionally improve 1.377 readability of certain aspects of reasoning, it is usually quite 1.378 redundant. Above, the basic proof outline gives already enough 1.379 structural clues for the system to infer both the rules and their 1.380 instances (by higher-order unification). Thus we may as well prune 1.381 - the text as follows. 1.382 -*} 1.383 + the text as follows. *} 1.384 1.385 lemma "(EX x. P (f x)) --> (EX y. P y)" 1.386 proof 1.387 @@ -399,12 +359,10 @@ 1.388 qed 1.389 qed 1.390 1.391 -text {* 1.392 - Explicit @{text \<exists>}-elimination as seen above can become quite 1.393 +text {* Explicit @{text \<exists>}-elimination as seen above can become quite 1.394 cumbersome in practice. The derived Isar language element 1.395 ``\isakeyword{obtain}'' provides a more handsome way to do 1.396 - generalized existence reasoning. 1.397 -*} 1.398 + generalized existence reasoning. *} 1.399 1.400 lemma "(EX x. P (f x)) --> (EX y. P y)" 1.401 proof 1.402 @@ -413,26 +371,21 @@ 1.403 then show "EX y. P y" .. 1.404 qed 1.405 1.406 -text {* 1.407 - Technically, \isakeyword{obtain} is similar to \isakeyword{fix} and 1.408 - \isakeyword{assume} together with a soundness proof of the 1.409 - elimination involved. Thus it behaves similar to any other forward 1.410 - proof element. Also note that due to the nature of general 1.411 - existence reasoning involved here, any result exported from the 1.412 - context of an \isakeyword{obtain} statement may \emph{not} refer to 1.413 - the parameters introduced there. 1.414 -*} 1.415 - 1.416 +text {* Technically, \isakeyword{obtain} is similar to 1.417 + \isakeyword{fix} and \isakeyword{assume} together with a soundness 1.418 + proof of the elimination involved. Thus it behaves similar to any 1.419 + other forward proof element. Also note that due to the nature of 1.420 + general existence reasoning involved here, any result exported from 1.421 + the context of an \isakeyword{obtain} statement may \emph{not} refer 1.422 + to the parameters introduced there. *} 1.423 1.424 1.425 subsubsection {* Deriving rules in Isabelle *} 1.426 1.427 -text {* 1.428 - We derive the conjunction elimination rule from the corresponding 1.429 - projections. The proof is quite straight-forward, since 1.430 - Isabelle/Isar supports non-atomic goals and assumptions fully 1.431 - transparently. 1.432 -*} 1.433 +text {* We derive the conjunction elimination rule from the 1.434 + corresponding projections. The proof is quite straight-forward, 1.435 + since Isabelle/Isar supports non-atomic goals and assumptions fully 1.436 + transparently. *} 1.437 1.438 theorem conjE: "A & B ==> (A ==> B ==> C) ==> C" 1.439 proof -